WORKS  OF 
FREDERICK  H.  GETMAN,  Ph.D. 

PUBLISHED    BY 

JOHN  WILEY  &  SONS,  Inc. 


Outlines  of  Theoretical  Chemistry. 

Small    8vo,    viii+ 467    pages,     104    figures. 
Cloth,  $3.50  net. 

An  Introduction  to  Physical  Science. 

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Laboratory  Exercises  in  Physical  Chemistry. 

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OUTLINES 


OF 


THEORETICAL  CHEMISTRY 


BY 

FREDERICK  H.  GETMAN,  Ph.D.  (Johns  Hopkins) 

11 

ASSOCIATE   PROFESSOR   OF   CHEMISTRY  IN 
BRYN   MAWR    COLLEGE 


FIRST  EDITION 

FIRST    THOUSAND 


NEW  YORK 

JOHN   WILEY   &   SONS,   INC. 

LONDON:  CHAPMAN  &  HALL,  LTD. 
1913 


COPYRIGHT,  1913, 
BY 

FREDERICK  H.  GETMAN 


Stanbope  jpress 

H.  GILSON    COMPANY 
BOSTON,  U.S.A. 


To 

«.  i.  «. 

WHOSE   TENDER  SYMPATHY  HAS  HELPED   TO 

LIGHTEN  THE  DARKNESS  OP  THE  DAYS 

DURING  WHICH  THESE  PAGES 

WERE  WRITTEN 


295802 


"I  look  upon  the  common  operations  and  practices  of  chymists,  almost 
as  I  do  on  the  letters  of  the  alphabet,  without  whose  knowledge  'tis  very 
hard  for  a  man  to  become  a  philosopher;  and  yet  that  knowledge  is  very  far 
from  being  sufficient  to  make  him  one." 

ROBERT  BOYLE.     (The  Sceptical  Chymist.) 


PREFACE. 


"  The  last  thing  that  we  find  in  making  a  book  is  to  know  what  we  must 
put  first."  —  PASCAL. 

THE  present  book  is  designed  to  meet  the  requirements  of 
classes  beginning  the  study  of  theoretical  or  physical  chemistry. 
A  working  knowledge  of  elementary  chemistry  and  physics  has 
been  presupposed  in  the  presentation  of  the  subject,  the  introduc- 
tory chapter  being  the  only  portion  of  the  book  in  which  space  is 
devoted  to  a  review  of  principles  with  which  the  student  is  assumed 
to  be  already  fairly  familiar.  With  the  exception  of  a  few  para- 
graphs in  which  the  application  of  the  calculus  is  unavoidable, 
no  use  is  made  of  the  higher  mathematics,  so  that  the  book  should 
be  intelligible  to  the  student  of  very  moderate  mathematical 
attainments.  Wherever  the  calculus  has  been  employed,  the 
student  who  is  unfamiliar  with  this  useful  tool  must  accept  the 
correctness  of  the  results  without  attempting  to  follow  the  suc- 
cessive operations  by  which  they  are  obtained. 

The  contributions  to  our  knowledge  in  the  domain  of  physical 
chemistry  have  increased  with  such  rapidity  within  recent  years, 
that  the  prospective  author  of  a  general  text  book  finds  himself 
confronted  with  the  vexing  problem  of  what  to  omit  rather  than 
what  to  include.  In  selecting  material  for  this  book,  the  author 
has  been  guided  in  large  measure  by  his  own  experience  in  teaching 
theoretical  chemistry  to  beginners  and  to  advanced  students. 
The  attempt  has  been  made  to  present  the  more  difficult  portions 
of  the  subject,  such  as  the  osmotic  theory  of  solutions,  the  laws 
of  equilibrium  and  chemical  action,  and  the  principles  of  electro- 
chemistry, in  a  clear  and  logical  manner.  While  the  treatment 
of  each  topic  is  necessarily  brief  yet  the  effort  has  been  made  to 
avoid  the  sacrifice  of  clearness  to  brevity. 

vii 


viii  PREFACE 

The  author  is  fully  convinced  from  his  own  experience  as  well 
as  from  that  of  his  colleagues,  that  the  complete  mastery  of  the 
fundamental  principles  of  the  science  is  best  attained  through 
the  solution  of  numerical  examples.  For  this  reason,  typical 
problems  have  been  appended  to  various  chapters  of  the  book. 

Numerous  references  to  original  papers  have  been  given  through- 
out, since  the  importance  of  literary  research  on  the  part  of  the 
student  is  conceded  by  all  teachers  to  be  of  prime  importance. 

While  a  brief  account  of  radioactive  phenomena  might  very 
properly  be  considered  to  lie  within  the  scope  of  a  general  outline 
of  theoretical  chemistry,  yet  owing  to  the  unparalleled  growth 
of  knowledge  in  this  field  during  the  last  decade,  the  author  has 
come  to  believe  that  a  condensed  statement  of  the  main  facts  of 
radiochemistry  would  not  be  of  sufficient  value  to  justify  the 
effort  involved  in  its  preparation. 

In  the  original  preparation  of  his  lectures,  and  in  their  evolution 
into  book  form,  the  author  has  had  frequent  occasion  to  consult 
Nernst's  "Theoretische  Chemie,"  Ostwald's  "Lehrbuch  der 
allgemeinen  Chemie,"  and  Van't  Hoff's  "Vorlesungen  ueber 
theoretische  und  physikalische  Chemie."  Among  other  books 
to  which  the  author  is  especially  indebted  are  the  following :  — 
Le  Blanc's  "Lehrbuch  der  Elektrochemie,"  Daneel's  "Elek- 
trochemie."  Text  books  of  Physical  Chemistry  edited  by  Sir 
William  Ramsay,  Bigelow's  "  Theoretical  and  Physical  Chem- 
istry." Jones'  "Elements  of  Physical  Chemistry,"  Reychler- 
Kuhn's  "  Physikalisch-chemische  Theorieen,"  and  Whetham's 
"Theory  of  Solution." 

In  the  preparation  of  the  problems  the  author  would  record  his 
indebtedness  to  Abegg  and  Sackur's  "Physikalisch-chemische 
Rechenauf gaben, "  and  to  Morgan's  "Elements  of  Physical 
Chemistry." 

It  is  a  pleasure  to  acknowledge  the  valuable  assistance  rendered 
by  Dr.  Eleanor  F.  Bliss  and  Dr.  Anna  Jonas,  who  have  read  and 
revised  the  proof  of  the  paragraphs  treating  of  crystalline  form. 
The  index  of  titles  and  names  has  been  prepared  by  the  author's 
wife  to  whose  untiring  patience  its  completeness  is  due.  The 
author  would  also  record  his  thanks  to  those  friends  whose  kindly 


PREFACE  ix 

criticism  has  helped  to  remove  many  blemishes.  Finally,  the 
author  would  express  his  appreciation  of  the  kindness  of  Messrs. 
Adam  Hilger  of  London,  and  Fritz  Koehler  of  Leipzig  who  have 
rendered  great  assistance  by  permitting  the  reproduction  of 
illustrations  of  apparatus  from  their  catalogs. 

FREDERICK  H.  GETMAN. 

STOCKBRIDGE,  MASS. 

Aug.  18,  1913. 


CONTENTS 

CHAPTER  PAGE 

PREFACE vii 

I.  Fundamental  Principles 1 

II.   Classification  of  the  Elements 20 

III.  The  Electron  Theory 34 

IV.  Gases 47 

V.   Liquids 79 

VI.  Solids 128 

VII.   Solutions 145 

VIII.   Dilute  Solutions  and  Osmotic  Pressure 165 

IX.   Association,  Dissociation  and  Solvation 203 

X.   Colloidal  Solutions 219 

XI.  Thermochemistry 236 

XII.  Homogeneous  Equilibrium 262 

XIII.  Heterogeneous  Equilibrium 278 

XIV.  Chemical  Kinetics 309 

XV.  Electrical  Conductance 335 

XVI.   Electrolytic  Equilibrium  and  Hydrolysis 372 

XVII.   Electromotive  Force 395 

XVIII.   Electrolysis  and  Polarization 432 

XIX.   Actinochemistry 444 

INDEX  OF  NAMES 457 

INDEX  OF  SUBJECTS 461 


THEORETICAL  CHEMISTRY. 


CHAPTER  I. 
FUNDAMENTAL  PRINCIPLES. 

Theoretical  Chemistry.  That  portion  of  the  science  of  chem- 
istry which  has  for  its  object  the  study  of  the  laws  controlling 
chemical  phenomena  is  called  theoretical  or  physical  chemistry. 
The  first  attempt  to  summarize  the  more  important  facts  and 
ideas  underlying  the  science  of  chemistry  was  made  by  Dalton  in 
1808  in  his  "New  System  of  Chemical  Philosophy."  The  birth 
of  the  science  of  theoretical  chemistry  may  be  considered  to  be 
coeval  with  the  appearance  of  Dalton's  epoch-making  book. 

Theoretical  chemistry  is  concerned  with  the  great  generaliza- 
tions of  chemical  science  and  bears  the  same  relation  to  chemistry 
that  philosophy  bears  to  the  whole  body  of  scientific  truth;  it 
aims  to  systematize  all  of  the  established  facts  of  chemistry 
and  to  discover  the  laws  governing  the  various  phenomena  of 
chemical  action. 

Law,  Hypothesis  and  Theory.  The  science  of  chemistry  is 
based  upon  experimentally  established  facts.  When  a  number 
of  facts  have  been  collected  and  classified  we  may  proceed  to  draw 
inferences  as  to  the  behavior  of  systems  under  conditions  which 
have  not  been  investigated.  This  process  of  reasoning  by  analogy 
we  term  generalization  and  the  conclusion  reached  we  call  a  law. 
It  is  apparent  that  a  law  is  not  an  expression  of  an  infallible  truth, 
but  it  is  rather  a  condensed  statement  of  facts  which  have  been 
discovered  by  experiment.  It  enables  us  to  predict  results  with- 
out recourse  to  experiment.  The  fewer  the  number  of  cases  in 
which  a  law  has  been  found  to  be  invalid,  the  greater  becomes  our 
confidence  in  it,  until  eventually  it  may  come  to  be  regarded  as 
tantamount  to  a  statement  of  fact. 

1 


2  THEORETICAL  CHEMISTRY 

Natural  laws  may  be  discovered  by  the  correlation  of  experi- 
mentally determined  facts,  as  outlined  above,  or  by  means  of  a 
speculation  as  to  the  probable  cause  of  the  phenomena  in  question. 
Such  a  speculation  in  regard  to  the  cause  of  a  phenomenon  is 
called  an  hypothesis. 

After  an  hypothesis  has  been  subjected  to  the  test  of  experiment 
and  has  been  shown  to  apply  to  a  large  number  of  closely  related 
phenomena  it  is  termed  a  theory. 

In  his  address  to  the  British  Association  (Dundee,  1912),  Profes- 
sor Senier  has  this  to  say :  "  While  the  method  of  discovery  in  chem- 
istry may  be  described  generally,  as  inductive,  still  all  the  modes  of 
inference  which  have  come  down  to  us  from  Aristotle,  analogical, 
inductive,  and  deductive,  are  freely  used.  An  hypothesis  is  framed 
which  is  then  tested,  directly  or  indirectly,  by  observation  and 
experiment.  All  the  skill,  and  all  the  resource  the  inquirer  can 
command,  are  brought  into  his  service;  his  work  must  be  accurate; 
and  with  unqualified  devotion  to  truth  he  abides  by  the  result, 
and  the  hypothesis  is  established,  and  becomes  a  part  of  the 
theory  of  science,  or  is  rejected  or  modified." 

Elements  and  Compounds.  All  definite  chemical  substances 
are  divided  into  two  classes,  elements  and  compounds. 

Robert  Boyle  was  the  first  to  make  this  distinction.  He  de- 
fined an  element  as  a  substance  which  is  incapable  of  resolution 
into  anything  simpler.  The  substances  formed  by  the  chemical 
combination  of  two  or  more  elements  he  termed  chemical  com- 
pounds. This  definition  of  an  element  as  given  by  Boyle  was 
later  proposed  by  Lavoisier  and,  notwithstanding  the  vast  accumu- 
lation of  scientific  knowledge  since  their  time,  the  definition  re- 
mains very  satisfactory  today. 

At  the  present  time  we  have  a  group  of  about  eighty  substances 
which  have  resisted  all  efforts  to  decompose  them  into  simpler 
substances.  These  are  the  so-called  chemical  elements.  It 
should  be  borne  in  mind,  however,  that  because  we  have  failed  to 
resolve  these  substances  into  simpler  forms  of  matter,  we  are  not 
warranted  in  maintaining  that  such  resolution  may  not  be  effected 
in  the  future. 

Recent  investigations  of  the  radioactive  elements  have  shown 


FUNDAMENTAL  PRINCIPLES  3 

that  they  are  continuously  undergoing  a  series  of  transformations, 
one  of  the  products  of  which  is  the  inactive  element  helium.  This 
behavior  is  contrary  to  the  old  view  that  transformation  of  one 
element  into  another  is  impossible.  At  first  the  attempt  was 
made  to  explain  it  by  assuming  that  the  radioactive  element  was 
a  compound  of  helium  with  another  element,  but  since  the  radio- 
active elements  possess  all  of  the  properties  characteristic  of 
elements  as  distinguished  from  compounds,  and  find  appropriate 
places  in  the  periodic  table  of  Mendeleeff,  the  "compound  theory" 
must  be  abandoned.  Uranium  and  thorium,  the  heaviest  ele- 
ments known,  appear  to  be  undergoing  a  process  of  spontaneous 
disintegration  over  which  we  have  no  control.  The  products 
of  this  disintegration  have  filled  the  gap  in  the  periodic  table 
between  thorium  and  lead  with  about  thirty  new  elements,  each 
of  which  is  in  turn  undergoing  transformations  similar  to  those 
of  the  parent  elements.  Professor  Soddy  *  says:  "In  spite  of 
the  existence  at  one  time  of  a  vague  belief  (  a  belief  which  has  no 
foundation),  that  all  matter  may  be  to  a  certain  extent  radioactive, 
just  as  all  matter  is  believed  to  be  to  a  certain  extent  magnetic,  it 
is  recognized  today  that  radioactivity  is  an  exceedingly  rare  prop- 
erty of  matter." 

Notwithstanding  these  remarkable  discoveries,  we  may  still  hold 
to  the  idea  of  an  element  as  suggested  by  Boyle  and  Lavoisier. 
Professor  Walker  says:  "The  elements  form  a  group  of  substances, 
singular  not  only  with  respect  to  the  resistance  which  they  offer 
to  decomposition,  but  also  with  respect  to  certain  regularities  dis- 
played by  them  and  not  shared  by  substances  which  are  designated 
as  compounds." 

Law  of  the  Conservation  of  Mass.  In  1774,  as  the  result  of 
a  series  of  experiments,  Lavoisier  established  the  law  of  the  con- 
servation of  mass  which  may  be  stated  as  follows :  In  a  chemical 
reaction  the  total  mass  of  the  reacting  substances  is  equal  to  the  total 
mass  of  the  products  of  the  reaction.  It  is  sometimes  stated  thus:  — 
the  total  mass  of  the  universe  is  a  constant;  but  this  form  of  state- 
ment is  open  to  the  objection  that  we  have  no  means  of  verification; 
it  is  a  statement  of  a  fact  which  transcends  our  experience. 
*  Chemistry  of  the  Radio-Elements,  p.  2. 


'  ^ 


4  THEORETICAL  CHEMISTRY 

Trie  law  of  the  conservation  of  mass  has  been  subjected  to  most 
rigid  investigation  by  Landolt  *  in  a  series  of  experiments  extend- 
ing over  a  period  of  fifteen  years. 

The  reacting  substances,  AB  and  CD, 
were  placed  in  the  two  arms  of  the  in- 
verted  U-tube  shown  in  Fig.  1  which  was 
then  sealed  at  S  and  the  whole  weighed 
upon  an  extremely  sensitive  balance.  The 
vessel  was  then  inverted  when  the  following 
reaction  took  place  :  — 

AB  +  CD-+AD  +  CB. 

When  the  reaction  was  complete  and  suf- 
ficient time  had  elapsed  to  allow  the  vessel 
AB|  ---  1  TZ=JCD  to  return  to  its  original  volume  (this  some- 
times required  nearly  three  weeks),  it  was 
weighed  again  using  every  precaution  to 
avoid  errors  and  any  gain  or  loss  in  weight 
noted. 


Fig.  1. 


Landolt  concluded  from  the  thirty  or  more  reactions  which  he 
studied  that  the  gain  or  loss  in  weight  was  less  than  one  ten- 
millionth  of  the  total  weight,  f 

Law  of  Definite  Proportions.  The  enunciation  of  the  law  of 
the  conservation  of  mass  and  the  introduction  of  the  balance  into 
the  chemical  laboratory  marked  the  beginning  of  a  new  era  in  the 
history  of  chemistry,  —  the  era  of  quantitative  chemistry.  As 
the  result  of  painstaking  experimental  work,  Richter  and  Proust 
announced  the  law  of  definite  proportions  about  the  beginning 
of  the  nineteenth  century.  This  law  may  be  expressed  thus: 
A  definite  chemical  compound  always  contains  the  same  elements 
united  in  the  same  proportion  by  weight. 

Shortly  after  the  enunciation  of  this  law  its  truth  was  questioned 
by  the  French  chemist  Berthollet.f  From  the  results  of  a  series 


*  Zeit.  phys.  chem.,  12,  1  (1893);  55>  589  (1906).   • 

t  An  excellent  summary  of  this  important  investigation  will  be  found  in 
the  Journal  de  Chimie  physique,  6,  625  (1908).  • 
t  Essai  de  statique  chimique  (1803). 


FUNDAMENTAL  PRINCIPLES  5 

of  brilliant  experiments,  he  became  convinced  that  chemical  reac- 
tions are  largely  controlled  by  the  relative  amounts  of  the  react- 
ing substances.  As  we  shall  see  later,  he  really  foreshadowed  the 
work  of  Guldberg  and  Waage  who  were  the  first  to  correctly  for- 
mulate the  influence  of  mass  on  a  chemical  reaction.  Berthollet 
argued  that  when  two  elements  unite  to  form  a  compound,  the 
proportion  of  one  of  the  elements  hi  the  compound  is  conditioned 
solely  by  the  amount  of  that  element  which  is  available.  This 
led  to  the  celebrated  controversy  between  Berthollet  and  Proust 
which  finally  resulted  in  the  establishment  of  the  latter's  original 
statement.  Subsequent  investigation  has  only  strengthened  our 
faith  in  the  law  of  definite  proportions. 

Law  of  Multiple  Proportions.  Elements  are  known  to  unite 
in  more  than  one  proportion  by  weight.  Dalton  analyzed  the 
two  compounds  of  carbon  and  hydrogen,  methane  and  ethylene, 
and  found  that  the  ratio  of  the  weights  of  carbon  to  hydrogen  in 
the  former  was  6 :  2  while  in  the  latter  it  was  6:1.  That  is,  for  the 
same  weight  of  carbon,  the  weights  of  the  hydrogen  in  the  two 
compounds  were  in  the  ratio  2:1. 

A  large  number  of  compounds  were  examined  and  similar 
simple  ratios  between  the  masses  of  the  constituent  elements 
were  found.  As  a  result  of  these  observations,  Dalton  *  formu- 
lated in  1808  the  law  of  multiple  proportions,  as  follows:  When 
two  elements  unite  in  more  than  one  proportion,  for  a  fixed  mass  of 
one  element  the  masses  of  the  other  element  bear  to  each  other  a  simple 
ratio.  Notwithstanding  the  fact  that  Dalton  was  a  careless 
experimenter  the  subsequent  investigations  of  Marignac  and 
others  have  established  the  validity  of  his  law. 

Law  of  Combining  Proportions.  Dalton  pointed  out  that  it  is 
possible  to  assign  to  every  element  a  definite  relative  weight  with 
which  it  enters  into  chemical  combination.  He  observed  that 
the  weights  or  simple  multiples  of  the  weights  of  the  different 
elements  which  unite  with  a  given  weight  of  a  definite  element, 
represent  the  weights  of  the  different  elements  which  combine 
with  each  other.  The  weights  of  the  elements  which  combine 
with  each  other  are  termed  their  combining  weights.  This  com- 
*  A  New  System  of  Chemical  Philosophy  (1808). 


6  THEORETICAL  CHEMISTRY 

prehensive  law  of  chemical  combination  may  be  stated  as  follows: 
Elements  combine  in  the  ratio  of  their  combining  weights  or  in 
simple  multiples  of  this  ratio.  It  will  be  observed  that  this  law 
really  includes  the  law  of  definite  and  the  law  of  multiple  pro- 
portions. 

If  we  assume  the  combining  weight  of  hydrogen  to  be  unity, 
the  combining  weights  of  chlorine,  oxygen  and  sulphur  will  be 
35.5,  8  and  16  respectively.  These  numbers  represent  the  ratios 
in  which  the  elements  substitute  each  other  in  chemical  com- 
pounds. Hydrochloric  acid,  for  example,  contains  35.5  parts  by 
weight  of  chlorine  to  1  part  by  weight  of  hydrogen  and  when 
oxygen  is  substituted  for  chlorine,  forming  water,  the  new  com- 
pound contains  8  parts  by  weight  of  oxygen  to  1  part  by  weight 
of  hydrogen.  Similarly,  if  the  oxygen  be  substituted  by  sulphur, 
forming  hydrogen  sulphide,  there  will  be  found  16  parts  by  weight 
of  sulphur  to  1  part  by  weight  of  hydrogen.  We  may  say,  then, 
that  35.5  parts  of  chlorine,  8  parts  of  oxygen  and  16  parts  of  sul- 
phur are  equivalent. 

A  chemical  equivalent  may  be  defined  as  the  weight  of  an  element 
which  is  necessary  to  combine  with  or  displace  1  part  by  weight 
of  hydrogen. 

The  Atomic  Theory.  In  very  early  times  two  different  views 
were  entertained  by  opposing  schools  of  Greek  philosophers  as  to 
the  mechanical  constitution  of  matter.  According  to  the  school 
of  Plato  and  Aristotle,  matter  was  thought  to  be  continuous 
within  the  space  it  appears  to  fill  and  to  be  capable  of  indefinite 
subdivision.  According  to  the  other  school,  first  taught  by 
Leucippus,  and  afterwards  by  Democritus  and  Epicurus,  matter 
was  considered  to  be  made  up  of  primordial,  extremely  minute 
particles,  distinct  and  separable  from  each  other  but  in  themselves 
incapable  of  division.  These  ultimate  particles  were  called  atoms 
(^aro/Mos),  signifying  something  indivisible.  While  the  Aristote- 
lian doctrine  held  sway  for  many  centuries  yet  the  notion  of  atoms 
was  revived  at  intervals.  Late  in  the  seventeenth  century,  Boyle 
seems  to  have  looked  upon  chemical  combination  as  the  result  of 
atomic  association. 

Guided  by  these  early  speculations  as  to  the  constitution  of 


FUNDAMENTAL  PRINCIPLES  7 

matter  and  influenced  by  his  study  of  the  writings  of  Sir  Isaac 
Newton,  Dal  ton  seems  to  have  formed  a  mental  picture  of  the 
part  played  by  atoms  in  the  act  of  chemical  combination.  After 
a  few  carelessly  performed  experiments,  the  results  of  which  ac- 
corded with  his  preconceived  ideas,  he  formulated  his  atomic 
theory. 

According  to  this  theory  matter  is  composed  of  extremely  mi- 
nute, indivisible  particles  or  atoms.  Atoms  of  the  same  element 
are  all  of  equal  weight,  but  atoms  of  different  elements  have 
weights  proportional  to  their  combining  numbers.  Chemical 
compounds  are  formed  by  the  union  of  atoms  of  different  kinds. 
This  theory  offers  a  simple,  rational  explanation  of  the  laws  of 
chemical  combination. 

Since  a  chemical  compound  results  from  the  union  of  atoms, 
each  of  which  has  a  definite  weight,  its  composition  must  be  in- 
variable, —  which  is  the  law  of  definite  proportions.  Again, 
when  atoms  combine  in  more  than  one  proportion,  for  a  fixed 
weight  of  atoms  of  one  kind,  the  weights  of  the  other  species  of 
atoms  must  bear  to  each  other  a  simple  ratio,  since  the  atoms  are 
indivisible  units.  This  is  clearly  the  law  of  multiple  proportions. 

Finally,  the  law  of  combining  weights  is  seen  to  follow  as  a 
necessary  consequence  of  the  atomic  theory,  since  the  experimen- 
tally determined  combining  weights  bear  a  simple  relation  to  the 
relative  weights  of  the  atoms. 

At  the  time  when  Dalton  proposed  his  atomic  theory,  the 
number  of  facts  to  be  explained  was  comparatively  small,  but 
with  the  enormous  growth  of  the  science  of  chemistry  during  the 
past  century  and  with  the  vast  accumulation  of  data,  the  theory 
has  proved  capable  of  affording  adequate  representation  of  all  of 
the  facts,  and  has  opened  the  way  to  many  important  generaliza- 
tions. 

While  the  atomic  theory  has  played  a  very  important  part  in  the 
development  of  modern  chemistry,  and  while  we  recognize  that  it 
helps  to  clarify  our  thinking  and  enables  us  to  construct  a  mental 
image  of  tiny  spheres  uniting  to  form  a  chemical  compound,  yet  we 
must  not  forget  the  fact  that  these  atoms  are  purely  hypothetical. 

Faraday  has  said:  "Whether  matter  be  atomic  or  not,  this 


8  THEORETICAL  CHEMISTRY 

much  is  certain,  that  granting  it  to  be  atomic,  it  would  appear 
as  it  now  does. "  Ostwald  believes  that  in  the  not  distant  future 
the  atomic  theory  will  be  abandoned  and  chemists  will  free  them- 
selves from  the  yoke  of  this  hypothesis,  relying  solely  upon  the 
results  of  experiment.  He  says:  "It  seems  as  if  the  adaptabil- 
ity of  the  atomic  hypothesis  is  near  exhaustion,  and  it  is  well 
to  realize  that,  according  to  the  lesson  repeatedly  taught  by  the 
history  of  science,  such  an  end  is  sooner  or  later  inevitable." 

Combining  Weights  and  Atomic  Weights.  The  problem  of 
determining  the  relative  atomic  weights  of  the  elements  would  at 
first  sight  appear  to  be  a  very  simple  matter.  This  might  appar- 
ently be  accomplished  by  selecting  one  element,  say  hydrogen, 
it  being  the  lightest  known  element,  as  the  standard;  a  compound 
of  hydrogen  and  another  element  may  then  be  analyzed  and  the 
amount  of  the  other  element  in  combination  with  one  part  by 
weight  of  hydrogen  determined.  This  weight  will  be  its  atomic 
weight  only  when  the  compound  contains  but  one  atom  of  each 
element.  To  determine  the  relative  atomic  weight,  therefore,  we 
must  know  in  addition  to  the  chemical  equivalent  of  the  element, 
the  number  of  atoms  present  in  the  compound.  For  example,  the 
analysis  of  water  shows  it  to  contain  8  parts  by  weight  of  oxygen 
to  1  part  by  weight  of  hydrogen;  the  chemical  equivalent  of 
oxygen  is,  therefore,  8,  and  if  water  contained  but  one  atom  of 
hydrogen  the  atomic  weight  of  oxygen  would  be  8.  It  can  be 
shown,  however,  that  water  contains  two  atoms  of  hydrogen 
and  one  atom  of  oxygen,  therefore,  the  atomic  weight  of  oxygen 
must  be  16.  It  is  evident,  therefore,  that  neither  the  analysis  nor 
the  synthesis  of  a  compound  is  sufficient  to  enable  us  to  determine 
the  number  of  atoms  of  an  element  combined  with  one  atom  of 
hydrogen.  We  shall  proceed  to  the  consideration  of  the  methods 
by  which  this  problem  may  be  solved. 

Gay-Lussac's  Law  of  Volumes.  Gay-Lussac  in  1808,  while 
studying  the  densities  of  gases  before  and  after  reaction,  announced 
the  following  law:  When  gases  combine  they  do  so  in  simple  ratios 
by  volume,  and  the  volume  of  the  gaseous  product  bears  a  simple 
ratio  to  the  volumes  of  the  reacting  gases  when  measured  under  like 
conditions  of  temperature  and  pressure.  Thus,  one  volume  of  hydro- 


FUNDAMENTAL  PRINCIPLES  9 

gen  combines  with  one  volume  of  chlorine  to  form  two  volumes 
of  hydrochloric  acid;  one  volume  of  oxygen  combines  with  two 
volumes  of  hydrogen  to  form  two  volumes  of  water  (vapor) ;  and 
one  volume  of  nitrogen  combines  with  three  volumes  of  hydrogen 
to  form  two  volumes  of  ammonia. 

In  a  previous  investigation,  Gay-Lussac  had  shown  that  all  gases 
behave  identically  when  subjected  to  changes  of  temperature  and 
pressure.  This  fact,  taken  together  with  the  simple  volumetric 
relation  just  enunciated  and  the  atomic  theory,  suggested  a  possible 
relation  between  the  number  of  ultimate  particles  in  equal  vol- 
umes of  different  gases. 

Berzelius  attempted  to  show  that  under  corresponding  condi- 
tions of  temperature  and  pressure,  equal  volumes  of  different 
gases  contain  the  same  number  of  atoms,  but  he  was  compelled 
to  abandon  the  assumption  as  untenable. 

Avogadro's  Hypothesis.  It  remained  for  the  Italian  physicist, 
Avogadro,*  in  1811,  to  point  out  the  distinction  between  atoms 
and  molecules,  terms  which  had  been  used  almost  synonymously 
up  to  his  time.  He  defined  the  atom  as  the  smallest  particle 
which  can  enter  into  chemical  combination,  whereas  the  molecule 
is  the  smallest  portion  of  matter  which  can  exist  in  a  free  state. 
He  then  formulated  the  following  hypothesis:!  Under  the  same 
conditions  of  temperature  and  pressure,  equal  volumes  of  all  gases 
contain,  the  same  number  of  molecules.  This  hypothesis  has  been 
subjected  to  such  rigid  experimental  and  mathematical  tests  that 
its  validity  cannot  be  questioned. 

Avogadro's  Hypothesis  and  Molecular  Weights.  According 
to  Gay-Lussac  when  hydrogen  and  chlorine  combine  to  form  hydro- 
chloric acid,  one  volume  of  hydrogen  unites  with  one  volume  of 
chlorine  yielding  two  volumes  of  hydrochloric  acid. 

According  to  the  hypothesis  of  Avogadro,  the  number  of  mole- 
cules of  hydrochloric  acid  is  double  the  number  of  molecules  of 
hydrogen  or  of  chlorine,  and,  consequently,  each  molecule  of  the 
reacting  gases  must  contain  at  least  two  atoms.  If  we  take 
hydrogen  as  the  unit  of  our  system  of  atomic  weights,  its  molec- 

*  Jour,  de  Phys.,  73,  58  (1811). 

t  Ampere  advanced  nearly  the  same  hypothesis  in  1814. 


10  THEORETICAL  CHEMISTRY 

ular  weight  must  be  2.  It  is  convenient  to  express  molecular 
and  atomic  weights  in  terms  of  the  same  unit,  for  then  the  molec- 
ular weight  of  a  substance  will  be  simply  the  sum  of  the  weights 
of  the  atoms  contained  in  the  molecule.  The  determination  of 
the  approximate  molecular  weight  of  a  substance,  therefore,  re- 
solves itself  into  ascertaining  the  mass  of  its  vapor  in  grams  which, 
under  the  same  conditions  of  temperature  and  pressure,  will 
occupy  the  same  volume  as  2  grams  of  hydrogen. 

This  weight  is  called  the  gram-molecular  weight  or  the  molar 
weight  of  the  substance,  while  the  corresponding  volume  is  known 
as  the  gram-molecular  or  molar  volume.  It  is  nearly  the  same  for 
all  gases  and  at  0°  and  760  mm.  it  may  be  taken  equal  to  22.4 
liters.  The  molecular  weights  obtained  from  vapor  density  meas- 
urements are  approximate  only,  because  of  the  failure  of  most 
gases  and  vapors  to  obey  the  simple  gas  laws,  a  condition  essen- 
tial to  the  strict  applicability  of  Avogadro's  hypothesis. 

Atomic  Weights  from  Molecular  Weights.  While  vapor 
density  determinations  as  ordinarily  carried  out  do  not  give  exact 
molecular  weights,  it  is  an  easy  matter  to  arrive  at  the  true  values 
when  we  take  into  consideration  the  results  of  chemical  analysis. 
It  is'  apparent  that  the  true  molecular  weight  must  be  the  sum  of 
the  weights  of  the  constituent  elements,  these  weights  being  exact 
multiples  or  submultiples  of  their  combining  proportions,  which 
proportions  have  been  determined  by  analysis  alone.  We  select, 
as  the  true  molecular  weight,  the  value  which  is  nearest  to  the 
approximate  molecular  weight  calculated  from  the  vapor  density 
of  the  substance.  For  example,  the  molecular  weight  of  ammonia, 
as  computed  from  its  vapor  density,  is  17.5  or,  in  other  words, 
17.5  grams  of  ammonia  occupy  the  same  volume  as  2  grams  of 
hydrogen,  measured  under  the  same  conditions  of  temperature 
and  pressure.  The  analysis  of  ammonia  shows  us  that  for  every 
gram  of  hydrogen,  there  are  present  4.67  grams  of  nitrogen. 
Hence  the  true  molecular  weight  must  contain  a  multiple  of  1  gram 
of  hydrogen  and  the  same  multiple  of  4.67  grams  of  nitrogen. 
The  problem  is,  to  find  what  integral  value  must  be  assigned  to 
x  in  the  expression,  x  (1  +  4.67),  in  order  that  it  may  give  the 
closest  approximation  to  17.5.  Clearly  if  x  =  3  the  value  of  the 


FUNDAMENTAL  PRINCIPLES 


11 


expression  becomes  17,  and  this  we  take  to  be  the  true  molecular 
weight.  This  gives  3  X  4.67  =  14  as  the  probable  atomic  weight 
of  nitrogen.  To  decide  whether  the  atomic  weight  of  nitrogen  is 
a  multiple  or  a  submultiple  of  14,  we  must  determine  the  molecu- 
lar weights  of  a  large  number  of  gaseous  or  vaporizable  compounds 
of  nitrogen  and  select  as  the  atomic  weight  the  smallest  quantity 
of  the  element  which  is  present  in  any  one  of  them. 

The  following  table  gives  a  list  of  seven  gaseous  compounds  of 
nitrogen  together  with  their  gram-molecular  weights,  and  the 
number  of  grams  of  the  element  in  the  gram-molecule. 


Compound. 

Gram-mol. 
Wt. 

Grams  Nitro- 
gen. 

Ammonia  

17 

14 

Nitric  oxide  

30 

14 

Nitrogen  peroxide  

46 

14 

Methyl  nitrate  

77 

14 

Cyanogen  chloride 

61  5 

14 

Nitrous  oxide 

44 

28 

Cyanogen  .... 

52 

28 

It  will  be  observed  that  the  least  weight  of  nitrogen  entering 
into  a  gram-molecular  weight  of  any  of  these  compounds  is  14 
grams,  and,  therefore,  we  accept  this  value  as  the  atomic  weight 
of  the  element,  although  there  is  still  a  very  slight  chance  that 
in  some  other  compound  of  nitrogen  a  smaller  weight  of  the  ele- 
ment may  be  found.  We  shall  proceed  to  point  out  that  there 
are  methods  by  which  the  probable  values  of  the  atomic  weights 
may  be  checked. 

Specific  Heat  and  Atomic  Weight.  In  1819  the  French  chem- 
ists, Dulong  and  Petit,*  pointed  out  a  very  simple  relation  between 
the  specific  heats  of  the  elements  in  the  solid  state  and  their 
atomic  weights.  This  relation,  known  as  the  law  of  Dulong  and 
Petit,  is  as  follows:  The  product  of  the  specific  heat  and  the  atomic 
weight  of  the  solid  elements  is  constant.  The  value  of  this  constant, 
called  the  atomic  heat,  is  approximately  6.4.  A  little  reflection 
will  show  that  an  alternative  statement  of  this  law  is  that  the 

*  Ann,  Chim.  Phys.,  10,  395  (1819). 


12 


THEORETICAL  CHEMISTRY 


atoms  of  the  elements  in  the  solid  state  have  the  same  thermal  capac- 
ity. The  specific  heats,  atomic  weights  and  atomic  heats  of 
several  elements  are  given  in  the  subjoined  table. 


Element. 

At.  Wt. 

Sp.  Ht. 

At.  Ht. 

Lithium 

7 

0  940 

6  6 

Glucinuna 

9 

0  410 

3  7 

Boron  (amorphous) 

11 

0  250 

2  8 

Carbon  (diamond) 

12 

0  140 

1  7 

Sodium  . 

23 

0  290 

6  7 

Silicon  (crystalline)  .  . 

28 

0  160 

4  5 

Potassium  

39 

0  166 

6  5 

Calcium  

40 

0  170 

6  8 

Iron  

56 

0  112 

6  3 

Copper 

63 

0  093 

5  9 

Zinc 

65 

0  093 

6  1 

Silver     . 

108 

0  056 

6  0 

Tin  

119 

0  054 

6  5 

Gold  

197 

0  032 

6  3 

Mercury  

200 

0  032 

6  4 

It  is  truly  remarkable  that  elements  differing  as  greatly  as 
lithium  and  mercury  differ,  not  only  in  atomic  weight  but  in 
other  properties  as  well,  should  have  identical  atomic  heats.  It 
will  be  observed  that  the  atomic  heats  of  boron,  silicon,  carbon 
and  glucinum  are  too  low.  This  departure  from  the  law  of  Dulong 
and  Petit  is  more  apparent  than  real,  for  in  the  statement  of  the 
law  there  is  no  specification  as  to  the  temperature  at  which  the 
specific  heat  should  be  determined.  The  specific  heats  of  all 
solids  vary  with  the  temperature,  this  variation  being  greater  in 
the  case  of  some  elements  than  in  that  of  others.  It  has  been 
shown  that  the  specific  heats  of  the  above  four  elements  increase 
rapidly  with  rise  of  temperature  and  approach  limiting  values. 
As  these  values  are  approached  the  product  of  specific  heat  and 
atomic  weight  approximates  more  and  more  closely  to  the  mean 
value  of  the  constant,  6.4. 

The  following  table  gives  the  values  obtained  by  Weber  *  for 
carbon  and  silicon. 


*  Pogg.  Ann.,  154,  367  (1875). 


FUNDAMENTAL  PRINCIPLES 


13 


CARBON   (DIAMOND). 


Temperature, 
degrees. 

Sp.  Ht. 

At.  Ht. 

-50 

0.0635 

0.76 

+10 

0.1128 

1.35 

85 

0.1765 

2.12 

206 

0.2733 

3.28 

607 

0.4408 

5.30 

806 

0.4489 

5.40 

985 

0.4589 

5.50 

CARBON   (GRAPHITE). 


Temperature, 
degrees. 

Sp.  Ht. 

At.  Ht. 

-50 

0.1138 

1.37 

+  10 

0.1604 

1.93 

61 

0.1990 

2.39 

202 

0.2966 

•       3.56 

642 

0.4454 

5.35 

822 

0.4539 

5.45 

978 

0.4670 

5.50 

SILICON. 


Temperature, 
degrees. 

Sp.  Ht. 

At.  Ht. 

-40 

0.136 

3.81 

+57 

0.183 

5.13 

129 

0.196 

5.50 

232 

0.203 

5.63 

It  is  evident  that  this  empirical  relation  can  be  used  to  deter- 
mine the  approximate  atomic  weight  of  an  element  when  its 
specific  heat  is  known,  thus 

.  ,  ,  6.4 

atomic  weight  =  r^ — ; — r- 

specific  heat 

The  law  of  Dulong  and  Petit  has  been  of  great  service  in  fixing 
and  checking  atomic  weights. 

About  twenty  years  after  the  law  of  Dulong  and  Petit  was 
formulated,  Neumann  *  showed  that  a  similar  relation  holds  for 
*  Pogg.  Ann.,  23,  1  (1831). 


14 


THEORETICAL  CHEMISTRY 


compounds  of  the  same  general  chemical  character.  Neumann's 
law  may  be  stated  thus:  Similarly  constituted  compounds  in  the 
solid  state  have  the  same  molecular  heat.  Subsequently  Kopp  * 
pointed  out  that  the  thermal  capacity  of  the  atoms  is  not  appreciably 
altered  when  they  enter  into  chemical  combination,  or  in  other  words, 
the  molecular  heat  of  solid  compounds  is  an  additive  property, 
being  made  up  of  the  atomic  heats  of  the  constituent  elements. 

For  example,  the  specific  heat  of  PbBr2  is  0.054  and  its  molec- 
ular weight  is  366.8,  therefore,  the  molecular  heat  is  0.054  X  366.8 
=  19.9.  Since  there  are  three  atoms  in  the  molecule,  19.9  -f-  3 
=  6.6  is  their  average  atomic  heat,  a  value  in  excellent  agree- 
ment with  the  constant  in  the  law  of  Dulong  and  Petit.  Neu- 
mann's law  may  be  used  to  estimate  the  atomic  heats  of  elements 
which  cannot  be  readily  investigated  in  the  solid  state.  The 
following  table  gives  a  list  of  atomic  heats  of  elements  in  the  solid 
state  derived  by  means  of  Neumann's  law. 


Element. 

At.  Ht. 

Element. 

At.  Ht. 

Hydrogen 

2  3 

Carbon 

1  8 

Oxygen 

4  0 

Silicon 

4  0 

Fluorine 

5  0 

Phosphorus 

5  4 

Nitrogen.  .  .  . 

5  5 

Sulphur 

5  4 

Isomorphism.  From  a  study  of  the  corresponding  salts  of 
phosphoric  and  arsenic  acids,  Mitscherlich  f  observed  that  they 
crystallize  with  the  same  number  of  molecules  of  water  and  are 
nearly  identical  in  crystalline  form,  it  being  possible  to  obtain 
mixed  crystals  from  solutions  containing  both  salts.  This  sug- 
gested to  Mitscherlich  a  line  of  investigation  which  resulted,  in 
1820,  in  the  establishment  of  the  law  of  isomorphism  which  bears 
his  name. 

This  law  may  be  stated  as  follows:  An  equal  number  of  atoms 
combined  in  the  same  manner  yield  the  same  crystal  form,  which  is 
independent  of  the  chemical  nature  of  the  atoms  and  dependent  upon 
their  number  and  position.  Thus,  when  one  element  replaces 

*  Lieb.  Ann.  (1864),  Suppl,  3,  5. 

t  Ann.  Chim.  Phys.  (2),  14,  172  (1820). 


FUNDAMENTAL  PRINCIPLES 


15 


another  in  a  compound  without  changing  its  crystalline  form, 
Mitscherlich  assumed  that  one  element  has  displaced  the  other, 
atom  for  atom.  For  example,  having  two  isomorphous  substances, 
such  as  BaCl2.2  H2O  and  BaBr2.2  H20,  we  assume  that  the  brom- 
ine in  the  second  compound  has  replaced  the  chlorine  in  the  first 
and,  if  the  atomic  weights  of  all  of  the  elements  in  the  first  com- 
pound are  known,  then  it  is  evident  that  the  atomic  weight  of  the 
bromine  in  the  second  compound  can  be  easily  calculated.  This 
method  was  largely  used  by  Berzelius  in  fixing  atomic  weights  and 
in  checking  the  values  obtained  by  the  volumetric  method.  It 
should  be  remembered  that  the  converse  of  the  law  of  isomorphism 
does  not  hold,  since  elements  may  replace  each  other,  atom  for 
atom,  without  preserving  the  same  form  of  crystallization.  Many 
exceptions  to  the  law  have  been  pointed  out.  For  example, 
Mitscherlich  himself  showed  that  Na^SC^  and  BaMn208  are  iso- 
morphous and  yet  the  two  molecules  do  not  contain  the  same 
number  of  atoms.  Furthermore,  careful  measurements  of  the 
interfacial  angles  of  crystals  have  revealed  the  fact  that  sub- 
stances which  have  been  regarded  as  isomorphous  are  only  approx- 
imately so,  thus  the  interfacial  angles  of  the  apparently  isomorph- 
ous crystalline  salts  given  in  the  following  table  differ  appreciably. 


Salt. 

Interfacial  Angle. 

MgSO4.7  H2O 
ZnSO4.7  H2O 
NiSO4.7  H2O 

89°  26' 

88°  53' 
88°  56' 

Ostwald  has  suggested  that  the  term  homeomorphous  be  applied  to 
designate  substances  which  have  nearly  identical  form.  At  best 
the  principle  of  isomorphism  is  only  an  approximation  and  should 
be  employed  with  caution. 

Valence.  During  the  latter  half  of  the  nineteenth  century 
the  usefulness  of  the  atomic  theory  was  greatly  enhanced  by  the 
introduction  of  certain  assumptions  concerning  the  combining 
power  of  the  atoms.  These  assumptions,  constituting  the  so- 
called  doctrine  of  valence,  were  forced  upon  chemists  in  order 
that  a  satisfactory  explanation  might  be  offered  of  the  phenomenon 


16  THEORETICAL  CHEMISTRY 

of  isomerism.  A  consideration  of  the  following  formulas,  — 
HC1,  H20,  NH3,  CH4,  —  shows  that  the  power  to  combine  with 
hydrogen  increases  regularly  from  chlorine,  which  combines  with 
hydrogen,  atom  for  atom,  to  carbon,  one  atom  of  which  is  capable 
of  combining  with  four  atoms  of  hydrogen.  Either  hydrogen  or 
chlorine,  each  of  which  is  capable  of  combining  with  but  one 
atom  of  the  other,  may  be  taken  as  an  example  of  the  simplest 
kind  of  atom.  Any  element  like  hydrogen  or  chlorine  is  called 
a  univalent  element,  whereas  elements  similar  to  oxygen,  nitrogen 
and  carbon,  which  are  capable  of  combining  with  two,  three  or 
four  atoms  of  hydrogen,  are  called  bivalent,  trivalent  and  quadri- 
valent elements  respectively.  Most  elements  belong  to  one  or 
the  other  of  these  four  classes,  although  quinquivalent,  sexivalent 
and  septivalent  elements  are  known.  The  familiar  bonds  or  link- 
ages of  structural  formulas  are  graphic  representations  of  the 
valence  of  the  atoms  constituting  the  molecule.  This  useful  con- 
ception of  valence  has  made  possible  the  prediction  of  the  prop- 
erties of  many  compounds  before  they  have  been  discovered  in 
nature  or  in  the  laboratory. 

Atomic  Weights.  Among  the  first  to  recognize  the  importance 
of  Dalton's  atomic  theory  was  the  Swedish  chemist,  Berzelius. 
He  foresaw  the  importance  for  chemists  of  a  table  of  exact  atomic 
weights  and  in  1810  he  undertook  the  task  of  determining  the 
combining  weights  of  most  of  the  known  elements.  For  nearly 
six  years  he  was  engaged  in  determining  the  exact  composition  of 
a  large  number  of  compounds  and  calculating  the  combining 
weights  of  their  constituent  elements,  thus  compiling  the  first 
table  of  atomic  weights.  • 

Numerous  investigators  since  Berzelius  have  been  engaged  in 
this  important  work,  among  whom  should  be  mentioned  Stas, 
Marignac,  Morley  and  Richards.  On  two  occasions  special  stimu- 
lus was  given  to  such  investigations.  The  first  occasion  was  in 
1815  when  Prout  suggested  that  the  atomic  weights  of  the  elements 
are  exact  multiples  of  the  atomic  weight  of  hydrogen.  The  values 
obtained  by  Berzelius  were  incompatible  with  the  hypothesis  of 
Prout,  although  the  atomic  weights  of  several  of  the  elements 
differed  but  little  from  integral  values.  To  test  the  accuracy  of 


FUNDAMENTAL  PRINCIPLES  17 

this  hypothesis,  Stas  undertook  the  determination  of  the  atomic 
weights  of  several  elements  with  a  degree  of  accuracy  such  that 
his  maximum  experimental  error  was  less  than  the  difference 
between  the  atomic  weight  found  and  the  nearest  whole  number. 
This  important  series  of  investigations  disproved  Prout's  hypoth- 
esis as  originally  stated.  The  second  occasion  when  the  investi- 
gation of  atomic  weights  received  a  special  impulse  was  in  1869 
when  Mendeleeff  brought  forward  the  periodic  classification  of  the 
elements.  When  the  elements  were  arranged  in  the  order  of  their 
atomic  weights,  several  of  them  were  found  to  fall  in  groups  with 
which  their  chemical  and  physical  properties  did  not  correspond, 
and  Mendeleeff  asserted  that  in  these  cases  the  commonly  accepted 
atomic  weights,  were  erroneous.  This  led  to  the  careful  redeter- 
mination  of  the  atomic  weights  which  Mendeleeff  had  asserted 
to  be  faulty,  and  in  most  cases  his  predictions  were  confirmed. 

In  connection  with  the  precise  determination  of  atomic  weights, 
the  work  of  Cannizzaro  in  the  latter  part  of  the  nineteenth  century 
should  be  mentioned.  He  emphasized  the  importance  of  Avoga- 
dro's  law  as  the  basis  of  atomic  weight  determinations,  and  drew 
a  sharp  distinction  between  atomic  and  molecular  weights,  thus 
bringing  order  out  of  confusion  and  rendering  possible  the  present 
system  of  atomic  weights.  In  recent  times,  the  most  noteworthy 
investigations  in  this  field  are  those  of  Morley  on  the  combining 
ratio  of  hydrogen  and  oxygen,  and  the  determination  by  T.  W. 
Richards  and  his  co-workers  of  the  atomic  weights  of  a  large 
number  of  elements. 

International  Atomic  Weights.  Dalton  selected  hydrogen, 
the  lightest  known  element,  as  the  unit  of  his  system  of  combin- 
ing weights  of  the  elements,  but  Berzelius  pointed  out  that  this 
was  an  unwise  choice  since  but  relatively  few  of  the  elements  form 
stable  compounds  with  hydrogen.  He  proposed,  therefore,  that 
oxygen  should  be  taken  as  the  standard,  assigning  to  it  the  arbi- 
trary value  100.  This  proposal  of  Berzelius,  to  substitute  oxygen 
for  hydrogen  as  the  unit  of  atomic  weights,  did  not  receive  serious 
consideration  until  quite  recently  when  the  International  Com- 
mittee on  Atomic  Weights  took  the  matter  up  and,  after  careful 
deliberation,  decided  in  favor  of  the  oxygen  standard.  The 


18 


THEORETICAL  CHEMISTRY 


1913. 
INTERNATIONAL  ATOMIC  WEIGHTS. 


Aluminium  
Antimony  
Argon 

Al 
Sb 
A 

O  =  16 

27.10 
120.20 
39.88 
74.96 
137.37 
208.00 
11.00 
79.92 
112.40 
132.81 
40.07 
12.00 
140.25 
35.46 
52.00 
58.97 
93.59 
63.57 
162.50 
167.70 
152.00 
19.00 
157.30 
69.90 
72.50 
9.10 
197.20 
3.99 
163.50 
1.008 
114.80 
126.92 
193.10 
55.84 
82.92 
139.00 
207.10 
6.94 
174.00 
24.32 
54.93 
200.60 

Molybdenum 

Mo 
Nd 
Ne 
Ni 
Nt 
N 
Os 
0 
Pd 
P 
Pt 
K 
Pr 
Ra 
Rh 
Rb 
Ru 
Sa 
Sc 
Se 
Si 
Ag 
Na 
Sr 
S 
Ta 
Te 
Tb 
Tl 
Th 
Tm 
Sn 
Ti 
W 
U 
V 
Xe 
Yb 
Yt 
Zn 
Zr 

O  =  16 

96.00 
144.30 
20.20 
58.68 
222.40 
14.01 
190.90 
16.00 
106.70 
31.04 
195.20 
39.10 
140.60 
226.40 
102.90 
85.45 
101.70 
150.40 
44.10 
79.20 
28.30 
107.88 
23.00 
87.63 
32.07 
181.50 
127.50 
159.20 
204.00 
232.40 
168.50 
119.00 
48.10 
184.00 
238.50 
51.00 
130.20 
172.00 
89.00 
65.37 
90.60 

Neodymium  .  .  . 

Neon.    ...           .       .    . 

Arsenic  

As 
Ba 
Bi 
B 
Br 
Cd 
Cs 
Ca 

r~\ 

Ce 
Cl 
Cr 

Co 
Cb 

Cu 

£ 

Eu 
F 
Gd 
Ga 
Ge 
Gl 
Au 
He 
Ho 
H 
In 
I 
Ir 
Fe 
Kr 
La 
Pb 
Li 
Lu 
Mg 
Mn 
Hg 

Nickel  
Niton  (radium  emanation). 
Nitrogen  

Barium 

Bismuth 

Boron 

Osmium  

Bromine 

Oxygen 

Cadmium  
Caesium  
Calcium  
Carbon  
Cerium  
Chlorine  
Chromium  
Cobalt 

Palladium 

Phosphorus 

Platinum     .         ... 

Potassium  . 

Praseodymium  

Radium  .  .  .  
Rhodium  

Rubidium  

Columbium 

Ruthenium  

Copper 

Samarium 

Dysprosium  
Erbium 

Scandium 

Selenium  
Silicon  
Silver                  

Europium 

Fluorine 

Gadolinium 

Sodium              

Gallium 

Strontium  

Germanium  
Glucinum 

Sulphur  

Tantalum  

Gold  
Helium  
Holmium 

Tellurium 

Terbium 

Thallium  

Hydrogen 

Thorium  

Indium 

Thulium                       

Iodine 

Tin                            

Iridium 

Titanium            

Iron 

Tungsten  

Krypton  
Lanthanum  
Lead  
Lithium  

Uranium  

Vanadium  

Xenon  

Ytterbium  (Neoytterbium) 
Yttrium                           

Lutecium  
M^agnesium 

Zinc                

IManganese 

Zirconium  

Mercury  

FUNDAMENTAL  PRINCIPLES  19 

atomic  weight  of  oxygen  is  taken  as  16,  and  the  unit  to  which  all 
atomic  weights  are  referred  is  one-sixteenth  of  this  weight.  The 
atomic  weight  of  hydrogen  on  this  basis  is  1.008.  Aside  from  the 
fact  that  most  of  the  elements  form  compounds  with  oxygen  which 
are  suitable  for  analysis,  the  atomic  weights  of  more  of  the  ele- 
ments approximate  to  integral  values  when  oxygen  instead  of  hy- 
drogen is  used  as  the  standard. 

The  table  on  page  18  gives  the  values  of  the  atomic  weights 
as  published  by  the  International  Committee  on  Atomic 
Weights  for  1913. 


CHAPTER  II. 
CLASSIFICATION  OF  THE  ELEMENTS. 

Early  Attempts  at  Classification.  Many  attempts  were  made 
to  classify  the  elements  according  to  various  properties,  such  as 
their  acidic  or  basic  characteristics  or  their  valence.  In  all  of 
these  systems  the  same  elements  frequently  found  a  place  in  more 
than  one  group,  and  elements  bearing  little  resemblance  to  each 
other  were  classed  together.  The  early  attempts  at  classifica- 
tion based  upon  the  atomic  weights  of  the  elements  were  not 
successful  owing  to  the  uncertainty  as  to  the  exact  numerical 
values  of  these  constants. 

Prout's  Hypothesis.  In  1815,  W.  Prout,  an  English  physician, 
observed  that  the  atomic  weights  of  the  elements,  as  then  given, 
did  not  differ  greatly  from  whole  numbers  when  hydrogen  was 
taken  as  the  standard.  Hence  he  advanced  the  hypothesis  that 
the  different  elements  are  polymers  of  hydrogen.  As  has  already 
been  pointed  out  this  hypothesis  led  Stas  to  undertake  his  refined 
determinations  of  the  atomic  weights  of  silver,  lithium,  sodium, 
potassium,  sulphur,  lead,  nitrogen  and  the  halogens.  As  a  result 
of  his  investigations  he  says:  "I  have  arrived  at  the  absolute 
conviction,  the  complete  certainty,  so  far  as  is  possible  for  a  human 
being  to  attain  to  certainty  in  such  matter,  that  the  law  of  Prout 
is  nothing  but  an  illusion,  a  mere  speculation  definitely  contra- 
dicted by  experience."  Notwithstanding  the  fact  that  Prout's 
hypothesis  as  originally  stated  was  thus  disproved  by  Stas,  it  still 
survived  in  a  modified  form  given  to  it  by  J.  B.  Dumas,  who  sug- 
gested that  one-half  of  the  atomic  weight  of  hydrogen  should  be 
taken  as  the  fundamental  unit.  When  Stas  showed  that  his 
experiments  excluded  this  possibility,  Dumas  suggested  that  the 
fundamental  unit  be  taken  as  one-quarter  of  the  atomic  weight 
of  hydrogen.  Having  begun  to  divide  and  subdivide,  there  was  no 
limit  to  the  process,  and  the  hypothesis  fell  into  disfavor,  although 

20 


CLASSIFICATION  OF  THE  ELEMENTS 


21 


the  belief  in  a  primal  element,  something  akin  to  the  protyle 
(•n-purr)  v\rj)  of  the  ancient  philosophers,  has  survived  and  in  modern 
times  has  reappeared  in  the  electron  theory.  \ 

Dobereiner's  Triads.  About  1817  J.  W.  Dobereiner*  observed 
that  groups  of  three  elements  could  be  selected  from  the  list  of  the 
elements,  all  of  which  are  chemically  similar,  and  having  atomic 
weights  such  that  the  atomic  weight  of  the  middle  member  is  the 
arithmetical  mean  of  the  first  and  third  members  of  the  group. 
These  groups  of  three  elements  he  termed  triads.  In  the  following 
table  a  few  of  these  triads  are  given. 


Element. 

At.  Wt. 

Mean  atomic 
weight  of 
triads. 

Lithium 

6  94 

Sodium 

23  00 

[     23  02 

Potassium  . 

39  10 

Calcium  

40.07 

Strontium 

87  63 

|     88  72 

Barium 

137  37 

Chlorine  

35.46 

Bromine  

79.92 

>     80.69 

Iodine 

126  92 

Sulphur  

32.07 

Selenium  

79.2 

>     78  78 

Tellurium  

127.5 

Phosphorus  

31.04 

Arsenic  

74.96 

75  62 

Antimony  

120.2 

This  simple  relation,  first  pointed  out  by  Dobereiner,  is  clearly 
a  foreshadowing  of  the  periodic  law. 

The  Helix  of  de  Chancourtois.  The  idea  of  arranging  the 
elements  in  the  order  of  their  atomic  weights  with  a  view,  to 
emphasizing  the  relationship  of  their  chemical  and  physical  prop- 
erties, seems  to  have  first  suggested  itself  to  M.  A.  E.  B.  de  Chan- 
courtois f  in  the  year  1862.  On  a  right-circular  cylinder  he  traced 


Ann.,  15,  301  (1825). 
t  Vis  Tellurique,  Classement  naturel  des  Corps  Simples. 


22 


THEORETICAL  CHEMISTRY 


what  he  termed  a  " telluric  helix"  at  a  constant  angle  of  45°  to  the 
axis.  On  this  curve  he  laid  off  lengths  corresponding  to  the 
atomic  weights  of  the  elements,  taking  as  a  unit  of  measure  a 
length  equal  to  one-sixteenth  of  a  complete  revolution  of  the 
cylinder.  He  then  called  attention  to  the  fact  that  elements  with 
analogous  properties  fall  on  vertical  lines  parallel  to  the  generatrix. 
Being  a  mathematician  and  a  geologist  he  did  not  express  himself 
in  such  terms  as  would  attract  the  attention  of  chemists  and  con- 
sequently his  work  remained  unnoticed  until  recent  times. 
The  Law  of  Octaves.  In  1864  J.  A.  R.  Newlands*  pointed 
out  that  if  the  elements  are  arranged  in  the  order  of  their  atomic 
weights,  the  eighth  element  has  properties  very  similar  to  the 
first;  the  ninth  to  the  second;  the  tenth  to  the  third;  and  so  on, 
or  to  employ  Newlands'  own  words :  "  The  eighth  element  starting  from 
a  given  one  is  a  kind  of  repetition  of  the  first,  like  the  eighth  note  of  an 
octave  in  music."  This  peculiar  relationship,  termed  by  New- 
lands  the  law  of  octaves,  is  brought  out  in  the  following  table. 


H 

Li 

Gl 

B 

C 

N 

O 

F 

Na 

Mg 

Al 

Si 

P 

S 

Cl 

K 

Ca 

Cr 

Ti 

Mn 

Fe 

Notwithstanding  the  fact  that  its  author  was  ridiculed  and  his 
paper  returned  to  him  as  unworthy  of  publication  in  the  proceed- 
ings of  the  Chemical  Society,  this  generalization  must  be  regarded 
as  the  immediate  forerunner  of  the  periodic  law. 

The  Periodic  Law.  Quite  independently  of  each  other  and 
apparently  in  ignorance  of  the  work  of  Newlands  and  de  Chan- 
courtois,  Mendeleeff  t  in  Russia  and  Lothar  Meyer  in  Germany, 
gained  a  far  deeper  insight  into  the  relations  existing  between 
the  properties  of  the  elements  and  their  atomic  weights.  In  1869, 
Mendeleeff  wrote:  —  "When  I  arranged  the  elements  according 
to  the  magnitude  of  their  atomic  weights,  beginning  with  the 
smallest,  it  became  evident  that  there  exists  a  kind  of  periodicity 


*  Chem.  News,  10,  94  (1864),  Ibid.,  12,  83  (1865). 
t  Lieb.  Ann.  Suppl.,  8,  133  (1874). 


CLASSIFICATION  OF  THE  ELEMENTS  23 

in  their  properties.  I  designate  by  the  name  'periodic  law'  the 
mutual  relations  between  the  properties  of  the  elements  and  their 
atomic  weights;  these  relations  are  applicable  to  all  the  elements 
and  have  the  nature  of  a  periodic  function."  This  important 
generalization  may  be  briefly  stated  thus:  The  properties  of  the 
elements  are  periodic  functions  of  their  atomic  weights. 

The  original  table  of  Mendeleeff  has  been  amended  and  modified 
as  new  data  has  accumulated  and  new  elements  have  been  dis- 
covered. The  accompanying  table,  though  containing  several 
new  elements  and  an  entirely  new  group,  is  essentially  the  same  as 
that  of  Mendeleeff.  It  consists  of  nine  vertical  columns,  called 
groups,  and  twelve  horizontal  rows  termed  series  or  periods.  The 
second  and  third  periods  contain  eight  elements  each,  and  are 
known  as  short  periods,  while  in  the  fourth  series,  starting  with 
argon,  it  is  necessary  to  pass  over  eighteen  elements  before  another 
element,  krypton,  is  encountered  which  bears  a  close  resemblance 
to  argon :  such  a  series  of  nineteen  elements  is  called  a  long  period. 
The  entire  table  is  composed  of  two  short  and  five  long  periods, 
the  last  one  being  incomplete.  The  positions  of  the  elements  are 
largely  determined  by  their  chemical  similarity  to  those  in  the 
same  group,  the  hyphens  indicating  the  positions  of  undiscovered 
elements.  The  elements  in  Group  VIII,  presented  difficulties 
when  Mendeleeff  attempted  to  place  them  according  to  their 
atomic  weights  and  so  he  was  obliged  to  group  them  by  themselves. 
This  group  has  wittily  been  designated  as  "the  hospital  for  incur- 
ables." An  examination  of  the  table  shows  that  the  valence  of 
the  elements  toward  oxygen  progresses  regularly  from  Group  O, 
containing  elements  which  exhibit  no  combining  power,  up  to 
Group  VIII,  where  it  attains  a  maximum  value  of  eight  in  the 
case  of  osmium.  The  valence  toward  hydrogen  on  the  other  hand 
increases  regularly  from  Group  VII  to  Group  IV  in  which  the 
elements  are  quadrivalent. 


24 


THEORETICAL  CHEMISTRY 


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CLASSIFICATION  OF  THE  ELEMENTS 


25 


n m ry 


THEORETICAL  CHEMISTRY 


Ether 

Hydrosphere 
Atmosphere 
Biosphere 


Lithosphere 


CLASSIFICATION  OF  THE  ELEMENTS  27 

The  formulas  of  the  typical  oxides  and  hydrides  of  the  elements 
in  the  several  groups  are  indicated  at  the  top  of  each  vertical 
column  in  the  table,  where  R  denotes  any  element  in  the  group. 
The  valence  of  elements  in  the  long  periods  are  apt  to  be  variable. 
The  elements  in  the  second  series  are  frequently  called  bridge 
elements,  since  they  bear  a  closer  relation  to  the  elements  in  the 
next  adjacent  group  than  they  do  to  any  other  members  of  the 
same  group  in  succeeding  series.  The  members  of  the  third  series 
are  styled  typical  elements,  because  they  exhibit  the  general  prop- 
erties and  characteristics  of  the  group.  Each  group  is  divided  into 
subgroups,  the  elements  on  the  right  and  left  sides  of  a  column 
forming  families,  the  members  of  which  are  more  closely  related 
than  are  all  of  the  elements  included  within  the  group.  In  other 
words  we  detect  a  kind  of  periodicity  within  each  group. 

In  any  given  series  the  element  with  lowest  atomic  weight 
possesses  the  strongest  basic  character.  Thus  we  find  the  strongly 
basic,  alkali  metals  on  the  left  side  of  the  fable,  while  on  the  right 
side  are  the  acidic  elements  such  as  the  halogens  and  sulphur. 

Graphic  Representation  of  the  Periodic  Law.  Of  the  numer- 
ous diagrams  which  have  been  devised  from  time  to  time  to  repre- 
sent the  periodic  relations  of  the  elements,  we  have  selected  two 
as  being  of  more  than  ordinary  interest.  The  diagram  shown  in 
Fig.  2,  is  a  modification  of  a  mode  of  representation  due  to  Sir 
William  Crookes.  The  values  of  the  atomic  weights  are  plotted 
on  a  vertical  scale,  on  the  right  and  left  of  which,  lines  representing 
valence  are  drawn,  univalent  elements  being  placed  on  line  I, 
bivalent  elements  on  line  II,  and  so  on.  The  curvature  of  these 
lines  is  intended  to  suggest  that  as  the  atomic  weight  increases 
the  chemical  activity  of  the  elements  diminishes.  Starting  with 
lithium  and  following  the  elements  in  the  order  of  their  atomic 
weights,  it  is  apparent  that  as  we  move  away  from  the  vertical 
axis  the  elements  are  positive,  whereas  when  we  approach  the 
axis  the  elements  acquire  a  negative  character.  In  Fig.  3,  we  have 
a  spatial  arrangement  of  the  elements,  due  to  Prof.  B.  K.  Emer- 
son,* in  which  the  elements  are  placed  on  a  helix  in  the  order  of 

*  Am.  Chem.  Jour.,  45,  160  (1911).  Lack  of  space  forbids  entering  into 
details  concerning  this  "Helix  Chemica,"  but  the  student  is  urged  to  consult 
Professor  Emerson's  original  paper. 


28 


THEORETICAL  CHEMISTRY 


their  atomic  weights  with  interspaces  equal  to  the  average  differ- 
ence of  the  atomic  weights  of  adjacent  elements. 

Periodicity  of  Physical  Properties.  Lothar  Meyer,  as  has  been 
pointed  out,  discovered  the  periodic  relations  of  the  elements  at 
about  the  same  time  as  Mendeleeff.  His  table  as  will  be  seen, 
differs  but  slightly  from  that  already  given.  In  this  table  the 


MEYER'S  TABLE. 


A  II  BAmBAIV  B 


AVIIB      vm 


—  L 

Li  7 

,  .    —    —  . 



019 

'  

•  , 

Bll 

—                    i—  

O12 

.  

—  , 

— 

016 

.  

F19 



.MaSS 

Mg  24 

_A127 

Si  28 

1  . 

P31 

832 

CI35 

—  

K39 

—  —  — 

Cu  63 

CaiO 
Zn65 

ScAl 
Ga  69 

TU8 
Ge72 

V51 

As  76 

Or  62 
Se78 

Mn  55 
Br80 


Fe56      Co  58     VIM 

Rb85 
Agio? 

Sr87 
Cdlll 

Y88 
In  113 

Zr90 
Sn  118_ 

Cb93 
Sbl20 

•  —      — 

Mo  96 
Tel26 

1126 

—  
RulOlRh  IQgpd  1Q6 

Cs  132 

Ba  136 

_  La  138 

C«U39 

- 

'   —  .  

•^  — 

.  ; 


Au  196 



E.166 



Hg  199 

Yb  172 

_ 

T1203 

1  



Pb205 

TalSl 



BI207 

•  

W  183 

Os  190    Ir  192  Ft  193 


•  —  .  — 

—  —  —  _ 

Th231 

•—  •  

U238 

•  

_ 

inclination  of  the  series  is  intended  to  represent  the  development 
of  the  thread  of  a  helix,  the  original  diagram  having  been  wrapped 
around  a  right-vertical  cylinder,  the  arrangement  being  much 
the  same  as  that  of  de  Chancourtois.  The  most  important  part 
of  Meyer's*  work,  however,  was  in  pointing  out  that  various 
physical  properties  of  the  elements  are  periodic  functions  of  their 
atomic  weights.  We  know  today  that  such  properties  as  specific 
gravity,  atomic  volume,  melting  point,  hardness,  ductility,  com- 
pressibility, thermal  conductivity,  coefficient  of  expansion,  specific 
refraction,  and  electrical  conductivity  are  all  periodic.  When  the 
numerical  values  of  these  properties  are  plotted  as  ordinates 
against  their  atomic  weights  as  abscissae,  we  obtain  wave-like 


Die  Modernen  Theorien  der  Chemie. 


CLASSIFICATION  OF  THE  ELEMENTS 


29 


curves  similar  to  those  shown  in  Fig.  4.  The  specific  heats  of 
the  elements  are  an  exception  to  the  general  rule.  According  to 
the  law  of  Dulong  and  Petit,  the  product  of  specific  heat  and 
atomic  weight  is  a  constant,  and  consequently  the  graphic  repre- 
sentation of  this  relation  must  be  an  equilateral  hyperbola. 

Applications  of  the  Periodic  Law.  Mendeleeff  pointed  out  the 
four  following  ways  in  which  the  periodic  law  could  be  employed: 
— (1)  The  classification  of  the  elements;  (2)  The  estimation  of  the 


FXiiion 
ialJgeu  Compoun 

V 


M 


Fig.  4. 


atomic  weights  of  elements;  (3)  The  prediction  of  the  properties 
of  undiscovered  elements;  and  (4)  The  correction  of  atomic 
weights. 

1.  Classification  of  Elements.    The  use  of  the  periodic  law  in 
this  direction  has  already  been  indicated.     It  is  without  doubt 
the  best  system  of  classification  known  and  is  to  be  ranked  among 
the  great  generalizations  of  the  science  of  chemistry. 

2.  Estimation   of  Atomic   Weights.    Because   of   experimental 
difficulties  it  is  not  always  possible  to  fix  the  atomic  weight  of  an 
element  by  determinations  of  the  vapor  densities  of  some  of  its 
compounds,  or  by  a  determination  of  its  specific  heat.     In  such 
cases  the  periodic  law  has  proved  of  great  value.     An  historic 


30  THEORETICAL  CHEMISTRY 

example  is  that  of  indium,  the  equivalent  weight  of  which  was 
found  by  Winkler  to  be  37.8.  The  atomic  weight  of  the  element 
was  thought  to  be  twice  the  equivalent  weight  or  75.6.  If  this 
were  the  correct  value  it  would  find  a  place  in  the  periodic  table 
between  arsenic  and  selenium.  Clearly  there  is  no  vacancy  in 
the  table  at  this  point  and  furthermore  its  properties  are  not 
allied  to  those  of  arsenic  or  selenium.  Mendeleeff  proposed  to 
assign  to  it  an  atomic  weight  three  times  its  equivalent  weight  or 
113.4,  when  it  would  fall  between  cadmium  and  tin  in  the  table. 
This  would  bring  it  in  the  same  group  with  aluminium,  the  typical 
element  of  the  group,  to  which  it  bears  a  close  resemblance.  This 
suggestion  of  Mendeleeff's  was  confirmed  by  a  subsequent  deter- 
mination of  the  specific  heat  of  indium. 

3.  Prediction  of  Properties  of  Undiscovered  Elements.  At  the 
time  when  Mendeleeff  published  his  first  table  there  were  many 
more  vacant  spaces  than  exist  in  the  present  periodic  table.  He 
ventured  to  predict  the  properties  of  many  of  these  unknown 
elements  by  means  of  the  average  properties  of  the  two  neighbor- 
ing elements  in  the  same  series,  and  the  two  neighboring  elements 
in  the  same  subgroup.  These  four  elements  he  termed  atomic 
analogues.  The  undiscovered  elements  Mendeleeff  designated  by 
prefixing  the  Sanskrit  numerals,  eka  (one),  dwi  (two),  tri  (three), 
and  so  on,  to  the  names  of  the  next  lower  elements  of  the  sub- 
group. When  the  first  periodic  table  was  published  there  were 
two  vacancies  in  Group  III,  the  missing  elements  being  called  by 
Mendeleeff  eka-aluminium  and  eka-boron,  while  in  Group  IV 
there  was  a  vacancy  below  titanium,  the  missing  element  being 
called  eka-silicon.  The  subsequent  discovery  of  gallium,  scandium 
and  germanium,  with  properties  nearly  identical  with  those  pre- 
dicted for  the  above  hypothetical  elements,  served  to  strengthen 
the  faith  of  chemists  in  the  periodic  law.  The  following  table 
illustrates  the  accuracy  of  Mendeleeff's  prognostications :  in  it  is 
given  a  comparison  of  a  few  of  the  properties  of  the  hypothetical 
element,  eka-silicon,  as  predicted  by  Mendeleeff  in  1871,  and  the 
corresponding  observed  properties  of  germanium,  discovered  by 
Winkler  fifteen  years  later. 


CLASSIFICATION  OF  THE  ELEMENTS 


31 


[Eka-silicon,  Es. 


Germanium,  Ge. 


Atomic  weight,  72. 

Specific  gravity,  5.5. 

Atomic  volume,  13. 

Metal  dirty  gray,   and  on  ignition 

yields  a  white  oxide,  EsO2. 
Element    decomposes    steam    with 

difficulty. 
Acids  have  slight  action,  alkalies  no 

pronounced  action. 


Action  of  Na  on  EsO2  or  on  EsK2F6 
gives  metal. 

The  oxide  EsO2  refractory. 

Specific  gravity  of  EsO2,  4.7. 

Basic  properties  of  EsO2  less  marked 
than  TiO2  and  SnO2,  but  greater 
than  SiO2. 

Forms  hydroxide  soluble  in  acids, 
and  the  solutions  readily  decom- 
pose forming  a  metahydrate. 

EsCl4  a  liquid  with  a  b.p.  below  100° 
and  a  sp.  gr.  of  1.9  at  0°. 

EsF4  not  gaseous. 

Es  forms  a  compound  Es(C2H5)4  boil- 
ing at  160°,  and  with  a  sp.  gr.  0.96. 


Atomic  weight,  72.3. 
Specific  gravity,  5.47. 
Atomic  volume,  13.2. 
Metal  grayish-white,  and  on  igni- 
tion yields  a  white  oxide,  GeO2. 
Element  does  not  decompose  water. 

Metal  not  attacked  by  HC1,   but 
acted  upon  by  aqua  regia. 

Solutions  of  KOH  have  no  action. 
Oxidized  by  fused  KOH. 

Ge  obtained  by  reduction  of  GeO2 
with  C,  or  of  GeK2F6  with  Na. 

The  oxide  GeO2  refractory. 

Specific  gravity  of  GeO2,  4.703. 
Basic  properties  of  GeO2  feeble. 


Acids  do  not  ppt.  the  hydroxide 
from  dil.  alkaline  solutions,  but 
from  cone,  solutions,  acids  ppt. 
GeO  or  a  metahydrate. 

GeCl4  a  liquid  with  a  b.p.  of  86°, 
and  a  sp.  gr.  at  18°  of  1.887. 

GeF4.3  H2O  a  white  solid. 

Ge  forms  a  compound  Ge(C2H6)4 
boiling  at  160°  and  with  a  sp.  gr. 
slightly  less  than  water. 


4.  Correction  of  Atomic  Weights.  When  an  element  falls  in  a 
position  in  the  periodic  table  where  it  clearly  does  not  belong, 
suspicion  as  to  the  correctness  of  its  atomic  weight  is  immediately 
aroused.  Frequently  a  redetermination  of  the  atomic  weight  has 
revealed  an  error  which,  when  corrected,  has  resulted  in  assigning 
the  element  to  a  place  among  its  analogues.  Formerly  the 
accepted  atomic  weights  of  osmium,  iridium,  platinum  and  gold 
were  in  the  order 

Os  >  Ir  >  Pt  >  Au. 

But  from  analogies  existing  between  osmium,  ruthenium  and  iron 
and  the  disposition  of  the  preceding  members  of  Group  VIII, 
Mendele"eff  predicted  that  the  atomic  weights  were  in  error  and 
that  the  order  of  the  elements  should  be 
Os  <  Ir  <  Pt  <  Au. 


32  THEORETICAL  CHEMISTRY 

Subsequent  atomic  weight  determinations  by  Seubert  substantiated 
Mendeleeff's  prediction. 

Defects  in  the  Periodic  Law.  While  the  arrangement  of  the 
elements  in  the  periodic  table  is  on  the  whole  very  satisfactory, 
there  are  several  serious  defects  in  the  system  which  should  be 
pointed  out.  At  the  very  outset  there  is  difficulty  in  finding  a 
place  for  hydrogen  in  the  system.  The  element  is  univalent  and 
falls  either  in  Group  I,  with  the  alkali  metals,  or  in  Group  VII  with 
the  halogens.  While  the  element  is  electro-positive  it  cannot  be 
considered  to  possess  metallic  properties.  It  forms  hydrides  with 
some  of  the  metallic  elements  and  can  be  displaced  by  the  halo- 
gens from  organic  compounds.  These  facts  make  it  extremely 
difficult  to  decide  whether  hydrogen  should  be  placed  in  Group  I 
or  Group  VII.  The  idea  has  been  advanced  that  hydrogen  is 
the  only  known  member  of  the  first  series  of  the  periodic  table. 

These  hypothetical  elements  have  been  styled  proto-elements, 
the  successive  members  of  the  series  being,  proto-glucinum,  proto- 
boron  and  so  on  to  the  last  element  in  the  series,  proto-fluorine.  To 
find  a  suitable  location  for  the  rare-earth  elements  in  the  periodic 
system  is  another  difficulty  which  has  not  been  satisfactorily  met. 
Brauner  considers  that  these  elements  should  all  be  grouped 
together  with  cerium  (at.  wt.  =  140.25),  but  owing  to  our  limited 
knowledge  of  the  properties  of  these  elements  it  seems  better  to 
defer  attempting  to  place  them  for  the  present.  In  the  group  of 
non-valent  elements  the  atomic  weight  of  argon  is  distinctly  higher 
than  that  of  potassium  in  the  next  group.  There  can  be  little 
doubt  that  the  values  of  the  atomic  weights  are  correct  and  i'-, 
is  evidently  impossible  to  interchange  the  positions  of  these  two 
elements  in  the  periodic  table,  since  argon  is  as  much  the  analogue 
of  the  rare  gases  as  potassium  is  of  the  alkali  metals.  A  similar 
discrepancy  occurs  with  the  elements,  tellurium  and  iodine.  The 
atomic  weight  of  the  former  element  is  appreciably  higher  than 
that  of  the  latter  and,  notwithstanding  the  attempts  of  numerous 
investigators  to  prove  tellurium  to  be  a  complex  of  two  or  more 
elements,  nothing  but  failure  has  attended  their  efforts.  Still 
another  anomaly  is  encountered  in  Group  VII,  where  manganese 
is  classed  with  the  halogen  family,  to  which  it  bears  much  less 


CLASSIFICATION  OF  THE  ELEMENTS  33 

resemblance  than  it  does  to  chromium  and  iron,  its  two  immediate 
neighbors. 

As  has  already  been  mentioned,  Group  VIII  is  made  up  of  non- 
conformable  elements.  If  the  properties  of  the  elements  are 
dependent  upon  their  atomic  weights,  it  should  be  impossible  for 
several  elements  having  almost  identical  atomic  weights  and 
different  properties  to  exist,  and  yet  such  is  the  case  with  the 
elements  of  Group  VIII.  The  elements  copper,  silver  and  gold, 
while  not  closely  resembling  the  other  members  of  Group  VIII, 
are  much  more  closely  allied  to  them  than  to  the  alkali  metals 
with  which  they  are  also  classed.  Notwithstanding  its  imper- 
fections, the  periodic  law  must  be  regarded  as  a  truly  wonderful 
generalization  which  future  investigations  will  undoubtedly  show 
to  be  but  a  fragment  of  a  more  comprehensive  law. 


CHAPTER  III. 
THE  ELECTRON  THEORY. 

Conduction  of  Electricity  through  Gases.  Within  recent 
years  the  discovery  of  new  facts  relative  to  the  conduction  of 
electricity  through  gases  has  led  to  the  development  of  the  so- 
called  electron  or  corpuscular  theory  of  matter.  Under  ordinary 
conditions  gases  are  practically  non-conductors  of  electricity,  but 
when  a  sufficiently  great  difference  of  potential  is  established 
between  two  points  within  a  gas  it  is  no  longer  able  to  withstand 
the  stress,  and  an  electric  discharge  takes  place  between  the  points. 
The  potential  necessary  to  produce  such  a  discharge  is  quite  high, 


*-€ 


To  pump 
Fig.  5 

several  thousand  volts  being  required  to  produce  a  spark  of  one 
centimeter  length  in  air  at  ordinary  pressures.  The  pressure  of 
the  gas  has  a  marked  effect  upon  the  character  of  the  discharge 
and  the  potential  required  to  produce  it.  If  we  make  use  of  a 
glass  vessel  similar  to  that  shown  in  Fig.  5,  the  effect  of  pressure 
on  the  nature  of  the  discharge  may  be  studied. 

This  apparatus  consists  of  a  straight  glass  tube  about  4  cm.  in 
diameter  and  40  cm.  long,  into  the  ends  of  which  platinum  elec- 
trodes are  sealed.  To  the  side  of  the  vessel  a  small  tube  is  sealed 
so  that  connection  may  be  established  with  an  air-pump  and 
manometer.  If  the  electrodes  are  connected  with  the  terminals 
of  an  induction  coil  and  the  pressure  within  the  tube  be  gradually 
diminished,  the  following  changes  in  the  character  of  the  dis- 

34 


THE  ELECTRON  THEORY 


35 


charge  will  be  observed.  At  first  the  spark  becomes  more  uni- 
form and  then  broadens  out,  assuming  a  bluish  color.  When  a 
pressure  of  about  0.5  mm.  is 
reached,  the  negative  electrode 
or  cathode  will  appear  to  be 
surrounded  by  a  thin  luminous 
layer;  next  to  this  will  be  a 
dark  region,  known  as  the 


Crookes'  dark  space;  adjoining 
this  will  be  a  luminous  portion,  . 
called  the  negative  glow,  and  beyond  this  will  be  seen  another  dark 
region  which  is  frequently  referred  to  as  the  Faraday  dark  space. 
Between  the  Faraday  dark  space  and  the  positive  electrode  or 
anode  is  a  luminous  portion,  called  the  positive  column.  By  a  slight 
variation  of  the  current  and  pressure  the  positive  column  can  be 
caused  to  break  up  into  alternate  light  and  dark  spaces  or  strice, 
the  appearance  of  which  is  dependent  upon  various  factors  such  as 


4- 


Fig.  7. 


the  nature  of  the  gas  and  the  size  of  the  tube.  If  we  use  a  modi- 
fication of  this  tube,  such  as  is  shown  in  Fig.  6,  and  diminish  the 
pressure  to  about  0.01  mm.,  a  new  phenomenon  will  be  observed. 
The  positive  column  will  vanish  and  the  walls  of  the  tube  opposite 
the  cathode  will  become  faintly  phosphorescent.  The  color  of 
the  phosphorescence  will  depend  upon  the  nature  of  the  glass: 
if  the  tube  is  made  of  soda  glass,  the  glow  will  be  greenish  yellow, 
while  with  lead  glass  the  phosphorescence  will  be  bluish.  The 
phosphorescence  is  due  to  the  bombardment  of  the  walls  of  the 


36  THEORETICAL  CHEMISTRY 

tube  by  very  minute  particles  projected  normally  from  the  cathode. 
These  streams  of  particles  are  called  the  cathode  rays. 

Some  Properties  of  Cathode  Rays.     The  following  are  among 
the  most  important  properties  of  the  cathode  rays :  — 

1.  The  cathode  rays  travel  in  straight  lines  normal  to  the  cathode: 
and  they  cast  shadows  of  opaque  objects  placed  in  their  path.     This 
property  may  be  demonstrated  by  means  of  the  apparatus  shown 
in  Fig.  7,  where  a  small  metallic  Maltese  cross  is  interposed  in  the 
path  of  the  rays,  a  distinct  shadow  being  cast  on  the  opposite  wall 
of  the  tube.     The  cross  may  be-  hinged  at  the  bottom  so  that 
it  can  be  dropped  out  of  the  path  of  the  rays,  when  the  usual 
phosphorescence  will  be  obtained. 

2.  The  cathode  rays  can  produce  mechanical  motion.    By  means 


1 


Fig.  8. 


of  the  apparatus  due  to  Sir  William  Grookes,  Fig.  8,  this  prop- 
erty of  the  cathode  rays  may  be  demonstrated.  Within  the 
vacuum  tube  is  placed  a  small  paddle  wheel  which  rolls  horizon- 
tally on  a  pair  of  glass  rails.  When  the  current  is  applied  to  the 
tube,  the  wheel  will  revolve,  moving  away  from  the  cathode.  By 
reversing  the  current,  the  wheel  will  stop  and  then  rotate  in  the 
opposite  direction  owing  to  the  reversal  of  the  direction  of  the 
catho.de  stream. 

3.  The  cathode  rays  cause  a  rise  of  temperature  in  objects  upon 
which  they  fall.  In  the  tube  shown  in  Fig.  9,  the  anode  consists 
of  a  small  piece  of  platinum :  this  is  placed  at  the  center  of  curva- 
ture of  the  spherical  cathode.  After  pumping  down  to  the 
proper  pressure,  if  a  strong  discharge  be  sent  through  the  tube, 


THE  ELECTRON  THEORY 


37 


the  anode  will  begin  to  glow,  and  if  the  action  of  the  current  be 
continued  long  enough,  the  platinum  plate  may  be  rendered  in- 
candescent, thus  showing  the 
marked  heating  effect  of  the 
cathode  rays. 

4.  Many  substances  become 
phosphorescent  on  exposure  to 
the  cathode  rays.  If  the  cathode 
rays  be  directed  upon  different 


r  To  pump 

substances,  such  as  calc-spar, 

barium  platino-cyanide,  willemite,  scheelite  and  various  kinds 
of  glass,  beautiful  phosphorescent  effects  may  be  observed.  This 
phosphorescent  property  is  useful  in  observing  and  experimenting 
with  the  cathode  rays. 

5.  The  cathode  rays  can  be  deflected  from  their  rectilinear  path  by 
a  magnetic  field.  In  studying  the  magnetic  deviation  of  the 
cathode  rays  a  tube  similar  to  that  shown  in  Fig.  10  has  been 
found  very  satisfactory.  An  aluminium  diaphragm,  A,  pierced 

LT 


Fig.  10. 

with  a  1  mm.  hole,  is  placed  in  front  of  the  cathode  while  at  the 
opposite  end  of  tube  is  placed  a  phosphorescent  screen,  D.  When 
the  discharge  takes  place  a  circular  phosphorescent  spot  will 
appear  on  D.  If  the  tube  be  placed  between  the  poles  of  an 
electromagnet,  the  phosphorescent  spot  will  move  at  right  angles 
to  the  direction  of  the  magnetic  field.  On  reversing  the  polarity 
of  the  magnet  the  spot  will  move  in  the  opposite  direction. 
Furthermore  the  direction  of  the  deflection  will  be  found  to  be 


38 


THEORETICAL  CHEMISTRY 


similar  to  that  produced  by  a  negative  charge  of  electricity  mov- 
ing in  the  same  direction  as  the  cathode  ray. 

6.  The  cathode  rays  can  be  deflected  from  their  rectilinear  path 
by  an  electrostatic  field.  The  same  tube  which  was  used  in  observ- 
ing the  magnetic  deflection  may  be  employed  in  studying  the 
effect  of  an  electrostatic  field.  Two  insulated  metal  plates,  B 
and  C}  are  placed  on  opposite  sides  of  the  tube  and  parallel  to 
each  other.  When  the  tube  is  in  action,  if  a  difference  of  poten- 
tial of  several  hundred  volts  be  applied  to  the  plates,  the  phos- 
phorescent spot  on  D  will  be  found  to  move,  the  direction  of  the 


Elect. 


Fig.  11. 

motion  being  the  same  as  that  of  a  negatively  charged  body  under 
the  influence  of  an  electrostatic  field.  Reversal  of  the  field  causes 
the  phosphorescent  spot  to  move  to  the  opposite  side  of  the 
screen. 

7.  The  cathode  rays  carry  a  negative  charge.  Probably  the  most 
important  characteristic  of  the  cathode  rays  is  their  ability  to 
carry  a  negative  charge.  While  the  magnetic  and  electrostatic 
deviation  of  the  rays  made  this  fact  more  than  probable,  it  re- 
mained for  Perrin  to  demonstrate  that  a  negative  electrification 
accompanies  the  cathode  stream.  A  modification  of  Perrin's 
apparatus  due  to  J.  J.  Thomson  is  shown  in  Fig.  11.  It  con- 


THE  ELECTRON  THEORY  39 

sists  of  a  spherical  bulb  to  which  is  sealed  a  smaller  bulb  and  a  long 
side  tube.  The  small  bulb  contains  the  cathode  C  and  the  anode 
A.  The  anode  consists  of  a  tight-fitting  brass  plug  pierced  by  a 
central  hole  of  small  diameter.  The  side  tube,  which  is  out  of 
the  direct  range  of  the  cathode  rays,  contains  two  coaxial  metallic 
cylinders  insulated  from  each  other,  each  being  perforated  with  a 
narrow  transverse  slit:  D  is  earth-connected  and  B  is  connected 
with  an  electrometer  by  means  of  the  rod  F.  When  the  tube  has 
been  pumped  down  to  the  proper  pressure  for  the  production  of 
cathode  rays,  a  phosphorescent  spot  will  appear  at  E  directly 
opposite  the  cathode  C.  Upon  testing  B  for  possible  electrifica- 
tion by  means  of  the  electrometer,  it  will  be  found  to  be  uncharged. 
If  the  cathode  stream  be  deflected  by  means  of  a  magnet  so  that 
the  rays  fall  upon  5,  a  sudden  charging  of  the  electrometer  will  be 
observed,  proving  that  B  is  becoming  electrified.  Upon  deflect- 
ing the  rays  still  further  so  that  they  are  no  longer  incident  upon 
J5,  the  accumulation  of  charge  will  immediately  cease.  If  the 
electrometer  be  tested  for  polarity,  it  will  be  found  to  be  negatively 
charged,  thus  proving  the  charge  carried  by  the  cathode  rays  to 
be  negative. 

8.  The  cathode  rays  can  penetrate  thin  sheets  of  metal.     In  1894 
Lenard  constructed  a  vacuum  tube  fitted  with  an  aluminium 
window  opposite  the  cathode.     He  showed  that  the  cathode  rays 
passed  through  the  aluminium  and  are  absorbed  by  different  sub- 
stances outside  of  the  tube,  the  absorption  varying  directly  with 
the  density  of  the  substance. 

9.  The  cathode  rays  when  directed  into  moist  air  cause  the  forma- 
tion of  fog.     This  phenomenon  has  been  shown  by  C.  T.  R.  Wilson 
to  be  due  to  the  minute  particles  in  the  cathode  stream  acting  as 
nuclei  upon  which  the  water  vapor  can  condense. 

Velocity  of  the  Cathode  Particle.  Since  the  cathode  rays 
consist  of  minute,  negatively-charged  particles  which  can  be 
deflected  by  a  magnetic  and  an  electrostatic  field,  it  is  possible 
to  measure  their  speed  and  to  compute  the  ratio  of  the  mass  of  a 
particle  to  its  charge.  The  special  form  of  tube  shown  in 
Fig.  12  was  devised  for  the  purpose  by  J.  J.  Thomson.  It  consists 
of  a  glass  tube  about  60  cm.  in  length,  furnished  with  a  flat  cir- 


40 


THEORETICAL  CHEMISTRY 


cular  cathode,  C,  and  an  anode,  A,  in  the  form  of  a  cylindrical 
brass  plug  about  2.5  cm.  in  length,  pierced  by  a  central  hole  1  mm. 
in  diameter.  Another  brass  plug,  B,  is  placed  about  5  cm.  away 
from  A,  the  two  holes  being  in  exactly  the  same  straight  line,  so 
that  a  very  narrow  bundle  of  rays  may  pass  along  the  axis  of  the 
tube  and  fall  upon  the  phosphorescent  screen  at  the  opposite 
end  of  the  tube.  Upon  this  screen  is  a  millimeter  scale,  SS'. 
Two  parallel  plates,  D  and  E,  are  sealed  into  the  tube  for  the  pur- 
pose of  establishing  an  electrostatic  field.  When  the  tube  is 
connected  with  an  induction  coil  or  other  source  of  high-potential, 
a  phosphorescent  spot  will  appear  at  F.  If  a  strong  magnetic 
field  be  applied,  the  lines  of  force  being  at  right  angles  to  the  plane 


Fig.  12. 

of  the  diagram,  the  rays  will  be  deflected  vertically  and  the  spot 
on  the  screen  will  move  from  F  to  G. 

Let  H  denote  the  strength  of  the  magnetic  field  and  let  m,  e 
and  v  represent  respectively  the  mass,  charge  and  velocity  of  a 
cathode  particle.  A  magnetic  field,  H,  acting  at  right  angles  to 
the  line  of  flight  of  the  cathode  particle  will  exert  a  force, 
HeVj  which  will  tend  to  deflect  the  particle  from  a  rectilinear 
path.  This  force  must  be  equal  to  the  centrifugal  force  of  the 
moving  particle  acting  outwards  along  its  radius  of  curvature. 
Therefore 

mv2 


or 


(D 


THE  ELECTRON  THEORY  41 


Since  H  and  r  can  both  be  measured,  the  ratio,  — ,  can  be  deter- 


mined. Now  if  a  difference  of  potential  be  established  between 
D  and  E,  and  the  lines  of  force  in  the  electrostatic  field  have  the 
proper  direction,  it  will  be  possible  to  alter  the  strength  of  the  field 
so  as  to  just  counterbalance  the  effect  of  the  magnetic  field,  and 
bring  the  phosphorescent  spot  back  to  F  again.  Under  these 
conditions,  if  X  denotes  the  strength  of  the  electrostatic  field,  we 
have 

Xe  =  Hev, 
or 

,-f.  :    .      |  (2) 

Since  X  and  H  can  both  be  measured,  v  can  be  calculated,  and  by 
introducing  the  value  so  obtained  into  equation  (1),  the  ratio  e/m 
can  be  evaluated.  By  this  method  the  average  value  of  v  has 
been  found  to  be  2.8  X  109  cm.  per  second,  while  1.7  X  107  is 
the  mean  value  of  a  large  number  of  determinations  of  the  ratio 
e/m. 

Comparison  of  the  Ratio  of  Charge  to  Mass  for  the  Cathode 
Particle  with  that  for  the  Ion  in  Electrolysis.  The  ratio  of  the 
charge  carried  by  an  ion  in  electrolysis  to  its  mass  can  be  easily 
computed.  Thus  it  may  be  shown  that  the  ratio  of  the  charge  E, 
of  the  hydrogen  ion  to  its  mass,  M,  in  electrolysis  is  about  1  X  104 

C.G.S.  units  or 

•pi 
jj  =  1  X  104  approximately. 

The  mass  of  the  hydrogen  ion  may  be  considered  to  be  identical 
with  that  of  the  hydrogen  atom,  the  lightest  atom  known.  Com- 
paring the  value  of  e/m  for  the  cathode  particle  with  the  value  of 
E/M  for  the  hydrogen  ion  in  electrolysis,  it  is  evident  that  the 
former  is  about  1700  times  greater  than  the  latter. 

Charge  Carried  by  the  Cathode  Particle.  Until  the  value 
of  the  charge  carried  by  the  cathode  particle  has  been  determined, 
it  is  clearly  impossible  to  compute  its  mass.  Thus,  if  we  consider 


42  THEORETICAL  CHEMISTRY 

the  last  statement  of  the  preceding  paragraph,  which  may  be 
formulated  as  follows  :  — 

e/m  :E/M  ::  1700  :  1, 

the  proportion  will  remain  unaltered  whether  ra  =  M/1700  and 
e  =  E,  or  e  =  1700  and  m  =  M.  The  method  employed  to  deter- 
mine the  charge  carried  by  a  cathode  particle  is  too  complicated 
for  a  detailed  description  in  this  place;  merely  the  general  outline 
will  be  given.  Upon  suddenly  expanding  a  volume  of  saturated 
water  vapor,  its  temperature  is  lowered,  and  a  cloud  forms,  each 
particle  of  dust  present  serving  as  a  nucleus  for  a  fog  particle.  If 
sufficient  time  be  allowed  for  the  mist  to  settle  and  the  vapor  to 
become  saturated  again,  a  repetition  of  the  preceding  process  will 
result  in  the  formation  of  less  mist,  owing  to  the  presence  of  fewer 
dust  particles.  By  repeating  the  operation  enough  times  the 
space  may  be  rendered  dust  free.  As  has  already  been  pointed 
out,  cathode  particles  serve  as  nuclei  for  the  condensation  of 
water  vapor,  their  function  being  similar  to  that  of  dust  particles. 
It  has  been  shown  by  Sir  George  G.  Stokes  that  if  a  drop  of  water 
of  radius  r,  be  allowed  to  fall  through  a  gas  of  viscosity  y,  then  the 
velocity  with  which  the  drop  falls  is  given  by  the  equation 


where  g  is  the  acceleration  due  to  gravity.  The  viscosity  of  air 
at  any  temperature  being  known,  a  cloud  can  be  produced  in  an 
appropriate  chamber  by  expansion  of  water  vapor  in  the  presence 
of  cathode  particles  and  the  speed,  v,  with  which  the  cloud  falls 
can  be  measured,  and  hence  r  can  be  calculated  by  means  of  equa- 
tion (3).  If  m  is  the  total  mass  of  the  cloud  and  n  is  the  number  of 
drops  per  cubic  centimeter,  then 

m  =  4/3  mrr3  (density  of  water  =1). 

From  a  simple  application  of  thermodynamics  m  may  be  deter- 
mined. Knowing  the  values  of  m  and  r,  the  number  of  drops 
in  the  cloud,  n,  which  is  the  same  as  the  number  of  cathode  par- 
ticles, can  be  calculated.  It  is  a  simple  matter  to  measure  the 
total  charge  in  the  expansion  chamber,  and  dividing  this  by  the 
total  number  of  charged  particles,  gives  the  charge  carried  by  a 


THE  ELECTRON  THEORY  43 

single  particle.  The  latest  determinations  of  J.  J.  Thomson  show 
this  to  be  3.4  X  10~10  electrostatic  units.  This  is  practically 
identical  with  the  calculated  value  of  the  charge  on  the  hydro- 
gen ion  in  electrolysis,  or  e  =  E  and  therefore  m  =  M/1700;  the 
mass  of  the  cathode  particle  is  1/1700  of  the  mass  of  the  atom 
of  hydrogen.  The  cathode  particle  has  the  smallest  mass  yet 
known  and  has  been  called  the  corpuscle  or  electron. 

Mass  and  Electric  Charge.  An  electrically-charged  body  in 
motion  has  an  apparent  mass  which  is  greater  than  its  real  mass. 
The  kinetic  energy  of  a  mass  is  expressed  by  the  familiar  equation 

K.E.  =  1/2  mv*. 

The  work  required  to  set  a  mass  in  motion  is  equivalent  to  the 
kinetic  energy  which  it  possesses  after  a  force  has  acted  upon  it. 
When  an  electrically  charged  body  is  set  in  motion  its  kinetic 
energy  is  augmented  by  the  energy  of  the  accompanying  magnetic 
field.  Thus  more  work  is  required  to  move  a  charged  than  an 
uncharged  body.  If  the  amount  of  work  required  to  set  a  definite 
mass  in  motion  be  denoted  by  TFi,  then  the  work  needed  when  the 
mass  has  acquired  an  electric  charge  will  be  represented  by  W\  +  TF2, 
where  Wz  is  the  work  required  to  move  the  charge  alone.  With 
masses  of  ordinary  size,  Wz  is  negligible  in  comparison  with  Wi, 
but  as  the  mass  becomes  smaller  and  its  speed  increases,  the  rela- 
tive importance  of  Wz  becomes  greater  until  eventually  with  a 
mass  as  small  as  the  corpuscle,  having  a  speed  approaching  that 
of  light,  the  value  of  W\  becomes  inappreciable.  That  is  equiva- 
lent to  saying  that  the  observed  mass  is  simply  electricity  in 
motion.  This  reasoning  has  been  tested  and  confirmed  by  Kauf- 
mann  who  determined  the  value  of  e/m  for  different  corpuscles. 
From  these  values  he  computed  the  masses  of  the  corpuscles,  and 
then  compared  these  masses  with  that  of  a  corpuscle  moving  with 
a  speed  such  that  Wz  is  negligible.  The  following  table  gives  the 
results  of  his  investigations,  the  first  column  showing  the  velocities, 
the  second  column  the  mass  found  in  terms  of  the  mass  of  the  slow- 
moving  corpuscle,  and  the  third  column  giving  the  values  of  the 
mass  calculated  on  the  assumption  that  it  is  wholly  due  to  a 
rapidly  moving  electric  charge, 


44 


THEORETICAL  CHEMISTRY 


Mass  found 

Velocity. 

(Mass  of  slow 
corpuscle  =1). 

Mass   Calculated. 

2.85  10cm./sec. 

3.09 

3.1 

2.7210        " 

2.43 

2.42 

2.59  10        " 

2.04 

2.0 

2.4810        " 

1.83 

1.66 

2.3610 

1.65 

1.5 

The  agreement  is  excellent  and  confirms  the  hypothesis  that  the 
entire  mass  of  a  corpuscle  is  due  to  the  motion  of  an  electric  charge. 
The  corpuscle  then,  is  not  a  minute,  hard,  inelastic  material  nucleus, 
but  is  simply  a  negative  electric  charge  in  motion.  The  term  elec- 
tron was  substituted  for  the  term  corpuscle  by  Stoney,  with  a  view 
to  indicating  its  electrical  character.  All  attempts  to  find  a  posi- 
tively charged  unit  have  thus  far  met  with  failure. 

The  Electronic  Hypothesis  of  Matter.  Notwithstanding  the 
fact  that  negative  electrons  have  been  produced  from  many 
different  gases  and  in  a  variety  of  ways,  up  to  the  present  time 
the  only  detectable  difference  between  them  is  in  their  velocity 
of  translation. 

Since  the  negative  electron  has  a  mass  much  smaller  than  that 
of  any  known  atom,  it  is  possible  that  it  may  be  a  constituent  of 
the  atoms  of  the  different  known  elements.  This  theory,  due  to 
J.  J.  Thomson  may  be  formulated  thus :  —  The  atoms  of  the  differ- 
ent elements  consist  of  an  assemblage  of  negative  electrons  held 
together  by  a  positively-charged  nucleus,  the  amount  of  positive  elec- 
trification being  equal  to  the  total  negative  charge  of  the  electrons,  thus 
rendering  the  atom  electrically  neutral.  The  possible  arrangements 
of  a  number  of  negatively-charged  particles  in  a  sphere  of  uni- 
form density  has  been  calculated  by  Thomson.  He  has  shown 
that  the  maximum  number  of  electrons  which  can  remain  in 
equilibrium  in  a  single  ring  is  five,  and  if  other  electrons  be  placed 
within  the  ring  a  larger  number  can  be  maintained  in  equilibrium 
in  one  ring.  For  example  a  ring  containing  six  electrons  would 
be  unstable,  but  if  a  seventh  electron  be  introduced  within  the 
six-membered  ring,  the  system  immediately  becomes  stable.  If 


THE  ELECTRON  THEORY 


45 


the  number  of  electrons  be  increased  they  will  arrange  themselves 
in  concentric  circles.  The  following  table  contains  the  results 
of  a  few  of  Thomson's  calculations  and  illustrates  the  necessary 
arrangement  of  electrons  in  concentric  rings  in  order  to  give 
stable  systems. 


Total  Number 
of  Electrons. 

Number  of  Electrons  in  Successive  Rings  Numbered  Outwards. 

1 

2 

3 

4 

5 

4 

5 
6 

7 
8 
9 
10 

4 
5 
5 

6 

7 
8 
8 

i 

1 

1 
1 
2 

59 
60 
61 
62 
63 
64 
65 
66 
67 

20 
20 
20 
20 
20 
20 
20 
20 
20 

16 
16 
16 
17 
17 
17 
17 
17 
17 

13 
13 
13 
13 
13 
13 
14 
14 
15 

8 
8 
9 
9 
10 
10 
10 
10 
10 

2 
3 
3 
3 
3 
4 
4 
5 
5 

These  results  are  in  excellent  agreement  with  the  experiments 
of  A.  M.  Mayer  on  floating  magnets.  He  thrust  uniformly  mag- 
netized needles  through  disks  of  cork,  and  then  floated  them  on 
the  surface  of  water  with  their  north  poles  submerged.  When 
the  north  pole  of  a  large  bar  magnet  was  brought  over  the  floating 
magnets  they  arranged  themselves  symmetrically,  the  form  of 
arrangement  depending  upon  the  number  of  magnets.  For  ex- 
ample, when  four  needles  were  used  they  arranged  themselves  at 
the  corners  of  a  square;  upon  adding  another  needle,  the  five 
needles  formed  a  regular  pentagon;  if  another  needle  was  added, 
it  moved  to  the  center  of  the  pentagon;  if  a  seventh  needle  was 
added  a  hexagon  was  formed  with  the  additional  needle  at  the 
center.  Thus  it  is  apparent  that  a  system  with  a  large  number 
of  magnets  on  the  outside  and  none  within  is  unstable.  This 


46  THEORETICAL  CHEMISTRY 

result  is  similar  to  that  obtained  mathematically  by  Thomson 
for  systems  of  electrons. 

In  terms  of  the  hypothesis  put  forward  by  Thomson,  the  atom 
consists  of  a  system  of  concentric  rings  of  negatively  charged  electrons 
grouped  within  a  sphere  of  positive  electrification.  Thus  the  hydrogen 
atom  is  assumed  to  consist  of  an  assemblage  of  1700  negatively 
charged  electrons  grouped  within  a  sphere  of  positive  electrifica- 
tion, the  surface  of  the  sphere  being  the  limiting  surface  of  the 
atom.  The  attendant  electrons  are  assumed  to  be  in  rapid  motion 
in  their  respective  orbits  about  the  center  of  the  sphere,  each 
being  subject  to  mutual  electronic  repulsion,  and  all  subject  to 
the  attraction  of  the  total  positive  charge.  The  system  may  be 
likened  to  a  planet  with  its  attendant  satellites.  This  theory 
furnishes  plausible  explanations  of  many  facts,  as  for  example, 
the  valence  of  the  elements:  —  An  atom  plus  or  minus  an  electron 
would  be  electrically  charged  and  capable  of  attracting  or  repel- 
ling other  charged  bodies;  an  atom  plus  or  minus  two  electrons 
would  possess  twice  the  power  of  attraction  or  repulsion  of  the 
first  atom.  In  other  words,  the  combining  capacity  of  the  second 
atom  is  twice  that  of  the  first,  or  the  first  atom  is  univalent  and 
the  second  bivalent.  The  electronic  hypothesis  furnishes  a  satis- 
factory explanation  of  optical  activity,  of  the  periodicity  of  the 
properties  of  the  elements  and  of  the  misfits  in  MendeleefFs 
table.  While  it  is  not  free  from  defects,  it  is  without  doubt  the 
best  attempt  which  has  been  made  to  shed  light  on  the  nature 
of  the  atom. 


CHAPTER  IV. 
GASES. 

The  Gas  Laws.  Matter  in  the  gaseous  state  possesses  the 
property  of  filling  completely  and  to  a  uniform  density  any  avail- 
able space.  Among  the  most  pronounced  characteristics  of 
gases  are  lack  of  definite  shape  or  volume,  low  density  and  small 
viscosity.  The  laws  expressing  the  behavior  of  gases  under  differ- 
ent conditions  are  relatively  simple  and  to  a  large  extent  are 
independent  of  the  nature  of  the  gas.  The  temperature  and 
pressure  coefficients  of  all  gases  are  very  nearly  the  same. 

In  1662,  Robert  Boyle  discovered  the  familiar  law  that  at 
constant  temperature,  the  volume  of  a  gas  is  inversely  proportional 
to  the  pressure  upon  it.  This  may  be  expressed  mathematically 
as  follows :  — 

v  oc  -  (temperature  constant) 

where  v  is  the  volume  and  p  the  pressure. 

In  1801,  Gay-Lussac  discovered  the  law  of  the  variation  of  the 
volume  of  a  gas  with  temperature. 

This  law  may  be  formulated  thus :  —  At  constant  pressure,  the 
volume  of  a  gas  is  directly  proportional  to  its  absolute  temperature, 
or 

v  oc  T  (pressure  constant). 

There  are  three  conditions  which  may  be  varied,  viz.,  volume, 
temperature  and  pressure.  The  preceding  laws  have  dealt  with 
the  relation  between  two  pairs  of  the  variables  when  the  third 
is  held  constant.  There  remains  to  consider  the  relation  between 
the  third  pair  of  variables,  pressure  and  temperature,  the  volume 
being  kept  constant.  Evidently  a  necessary  corollary  of  the  first 
two  laws  is  that  at  constant  volume,  the  pressure  of  a  gas  is  directly 
proportional  to  its  absolute  temperature,  or 

p  oc  T  (volume  constant). 
47 


48  THEORETICAL  CHEMISTRY 

These  three  laws  may  be  combined  into  a  single  mathematical 
expression  as  follows  :  — 

v  oc  -  (T  const.)  law  of  Boyle, 

v  QC  T  (p  const.)  law  of  Gay-Lussac; 
combining  these  two  variations  we  have, 

T 

V  OC  —  , 
P 

or  introducing  a  proportionality  factor  k, 

/  T 

v  =  k  —  > 

P 
or 

vp  =  kT.  (1) 

If  the  temperature  of  the  gas  be  0°  (273°  absolute),  and  the  corre- 
sponding volume  and  pressure  VQ  and  PQ  respectively,  then  (1) 
becomes 

v0p0  =  273  fc, 

and 


~273* 

eliminating  the  constant  k  between  (1)  and  (2),  we  have 


For  any  one  gas  the  term        is  a  constant.     If  VQ  is  the  volume 


of  1  gram  of  gas  at  0°  and  76  cm.,  we  write 

vp  =  rT,  (3) 

where  v  is  the  volume  of  1  gram  of  gas  at  the  temperature  T  and 
the  pressure  p,  and  r  is  a  constant  called  the  specific  gas  constant. 
On  the  other  hand  when  VQ  denotes  the  volume  of  one  mol.  of  gas 
at  0°  and  76  cm.  (22.4  liters),  the  equation  becomes 

vp  =  RT,  (4) 

where  R  is  termed  the  molecular  gas  constant,  which  has  the  same 


GASES  49 

value  for  all  gases.     If  M  is  the  molecular  weight  of  the  gas,  Mr 
=  R.     Equation  (4)  is  the  fundamental  gas  equation. 

Evaluation  of  the  Molecular  Gas  Constant.  Since  the  product 
of  p  and  v  represents  work,  and  T  is  a  pure  number,  R  must  be 
expressed  in  energy  units.  There  are  four  different  units  in  which 
the  molecular  gas  constant  is  commonly  expressed,  viz.,  (1)  gram- 
centimeters,  (2)  ergs,  (3)  calories,  and  (4)  liter-atmospheres. 

1.  R  in  gram-centimeters.    The  volume,  v,  of  1  mol.  of  gas  at 
0°  and  76  cm.  is  22.4  liters  or  22,400  cc.     The  pressure,  p,  is  76  cm. 
multiplied  by  13.59,  (the  density  of  mercury),  or  1033.3  grams  per 
square  centimeter.     Substituting  we  obtain 

_  po*  _  1033.3  X  22,400 
K  -  —jjr  —  273  —          ~   •  J£*u  ^      m* 

2.  R  in  ergs.     To  convert  gram-centimeters  into  ergs  we  must 
multiply  by  the  acceleration  due  to  gravity,  g   =  980.6  cm.  per 
sec.   per  sec.,  or 

R  =  84,760  X  980.6  =  83,150,000  ergs. 

3.  R  in  calories.    To  express  work  in  terms  of  heat,  we  must 
divide  by  the  mechanical  equivalent  of  heat,  or  since  1  calorie  is 
equivalent  to  42,640  gr.  cm.  or  41,830,000  ergs,  we  have 

D      83,150,000         nn         f 
=  41830000  =  (approximately  2  cal.) 

4.  R  in  liter-atmospheres.    A  liter-atmosphere  may  be  defined 
as  the  work  done  by  1  atmosphere  on  a  square  decimeter  through 
a  decimeter.     If  pQ  is  the  pressure  in  atmospheres,  and  VQ  is  the 
volume  in  liters,  we  have 


R  =       ?  =          =0.0821  liter-atmosphere. 

1 


Deviations  from  the  Gas  Laws.  Careful  experiments  by 
Amagat  *  and  others  on  the  behavior  of  gases  over  extended  ranges 
of  temperature  and  pressure  have  shown  that  the  fundamental 
gas  equation,  pv  =  RT,  is  not  strictly  applicable  to  any  one  gas, 
the  deviations  depending  upon  the  nature  of  the  gas  and  the 
conditions  under  which  it  is  observed.  It  has  been  shown  that 
the  gas  laws  are  more  nearly  obeyed  the  lower  the  pressure,  the 
*  Ann.  Chim.  phys.  (5)  19,  345  (1880). 


50 


THEORETICAL  CHEMISTRY 


higher  the  temperature  and  the  further  the  gas  is  removed  from 
the  critical  state.  A  gas  which  would  conform  to  the  require- 
ments of  the  fundamental  gas  equation  is  called  an  ideal  or  per- 
fect gas.  Almost  all  gases  are  far  from  ideal  in  their  behavior. 
At  constant  temperature  the  product,  pv,  in  the  gas  equation  is 
constant,  so  that  if  we  plot  pressures  as  abscissae  and  the  corre- 


P  (atmospheres) 
Fig.  13. 

sponding  values  of  pv  as  ordinates,  for  an  ideal  gas  we  should 
obtain  a  straight  line  parallel  to  the  axis  of  abscissae,  as  shown  in 
Fig.  13.  The  results  obtained  by  Amagat  with  three  typical 
gases  are  also  shown  in  the  same  diagram.  It  will  be  apparent 
that  all  of  these  gases  depart  widely  from  ideal  behavior.  In 
the  case  of  hydrogen  pv  increases  continuously  with  the  pressure, 


GASES  51 

while  with  nitrogen  and  carbon  dioxide  it  first  decreases,  attains 
to  a  minimum  value  and  beyond  that  point  increases  with  increas- 
ing pressure.  With  the  exception  of  hydrogen,  all  gases  show  a 
minimum  hi  the  curve,  thus  indicating  that  at  first  the  compress- 
ibility is  greater  than  corresponds  with  the  law  of  Boyle,  but 
diminishes  continuously  until,  for  a  short  range  of  pressure,  the  law 
is  followed  strictly:  beyond  this  point  the  compressibility  is  less 
than  Boyle's  law  requires. 

Hydrogen  is  exceptional  in  that  it  is  always  less  compressible 
than  the  law  demands.  This  is  true  for  all  ordinary  tempera- 
tures, but  it  is  highly  probable  that  at  extremely  low  temperatures 
the  curve  would  show  a  minimum.  The  two  curves  for  carbon 
dioxide  at  31°.5  and  100°  illustrate  the  fact  that  the  deviations 
from  the  gas  laws  become  less  as  the  temperature  increases.  The 
deviations  of  gases  from  the  laws  of  Boyle  and  Gay-Lussac,  as  well 
as  their  behavior  in  general,  may  be  satisfactorily  accounted  for 
on  the  basis  of  the  kinetic  theory. 

Kinetic  Theory  of  Gases.  The  first  attempt  to  explain  the 
properties  of  gases  on  a  purely  mechanical  basis  was  made  by 
Bernoulli  in  1738.  Subsequently,  through  the  labors  of  Kroenig, 
Clausius,  Maxwell,  Boltzmann  and  others,  his  ideas  were  developed 
into  what  is  known  today  as  the  kinetic  theory  of  gases.  Accord- 
ing to  this  theory,  gases  are  considered  to  be  made  up  of  minute, 
perfectly  elastic  particles  which  are  ceaselessly  moving  about 
with  high  velocities,  colliding  with  each  other  and  with  the  walls 
of  the  containing  vessel.  These  particles  are  identical  with  the 
molecules  defined  by  Avogadro.  The  volume  actually  occupied 
by  the  gas  molecules  is  supposed  to  be  much  smaller  than  the 
volume  filled  by  them  under  ordinary  conditions,  thus  allowing 
the  molecules  to  move  about  free  from  one  another's  influence 
except  when  they  collide.  The  distance  through  which  a  molecule 
moves  before  colliding  with  another  molecule  is  known  as  its  mean 
free  path.  In  terms  of  this  theory,  the  pressure  exerted  by  a  gas 
is  due  to  the  combined  effect  of  the  impacts  of  the  moving  molecules 
upon  the  walls  of  the  containing  vessel,  the  magnitude  of  the 
pressure  being  dependent  upon  the  kinetic  energy  of  the  mole- 
cules and  their  number. 


52 


THEORETICAL  CHEMISTRY 


Derivation  of  the  Kinetic  Equation.  Starting  with  the  assump- 
tions already  made,  it  is  possible  to  derive  a  formula  by  means  of 
which  the  gas  laws  may  be  deduced.  Imagine  n  molecules,  each 
having  a  mass,  m,  confined  within  the  cubical  vessel  shown  in 
Fig.  14,  the  edge  of  which  has  a  length,  I.  While  the  different 
molecules  are  doubtless  moving  with  different  velocities,  there 
must  be  an  average  velocity  for  all  of  them.  Let  c  denote  this 
mean  velocity  of  translation.  The  molecules  will  impinge  upon 
the  walls  in  all  directions  but  the  velocity  of  each  may  be  resolved 
according  to  the  well-known  dynamical  principle  into  three  corn- 


Fig.  14. 

ponents,  x,  y  and  2,  parallel  to  the  three  rectangular  axes,  X,  Y 
and  Z.  The  analytical  expression  for  the  velocity  of  a  single 
molecule,  M,  is 


In  words,  this  means  that  the  effect  of  the  collision  of  the  molecule 
upon  the  wall  of  the  containing  vessel,  is  equivalent  to  the  com- 
bined effect  of  successive  collisions  of  the  molecule  perpendicular 
.to  the  three  walls  of  the  cubical  vessel  with  the  velocities  x,  y  and 
z  respectively.  Fixing  our  attention  upon  the  horizontal  com- 
ponent, the  molecule  will  collide  with  the  wall  with  a  velocity  x, 
and  owing  to  its  perfect  elasticity  it  will  rebound  with  a  velocity 


GASES  53 

—  x,  having  suffered  no  loss  in  kinetic  energy.  The  momentum 
before  collision  was  mx  and  after  collision  it  will  be  —  mx,  the 
total  change  in  momentum  being  2  mx.  The  distance  between 
the  two  walls  being  I,  the  number  of  collisions  on  a  wall  in  unit 
time  will  be,  x/l,  and  the  total  effect  of  a  single  molecule  in  one 
direction  in  unit  time  will  be  2  mx  •  x/l  =  2  mx2/l.  The  same 
reasoning  is  applicable  to  the  other  components,  so  that  the  com- 
bined action  of  a  single  molecule  on  the  six  sides  of  the  vessel 
will  be 


There  being  n  molecules,  the  total  effect  will  be  —  -,  --     The 

entire  inner  surface  of  the  cubical  vessel  being  6  I2,  the  pressure  p, 
on  unit  area,  will  be 

2  mnc2      1   mnc2  . 


but  since  I3  is  the  volume  of  the  cube,  which  we  will  denote  by  v, 
we  have 

1   mnc2 

o       v 

or 

i  $ , 

pv  =  -•mnc2. 

This  is  the  fundamental  equation  of  the  kinetic  theory  of  gases. 
While  the  equation  has  been  derived  for  a  cubical  vessel,  it  is 
equally  applicable  to  a  vessel  of  any  shape  whatever,  since  the 
total  volume  may  be  considered  to  be  made  up  of  a  large  number 
of  infinitesimally  small  cubes,  for  each  of  which  the  equation  holds. 
Deductions  from  the  Kinetic  Equation.  Law  of  Boyle.  In 
the  fundamental  kinetic  equation,  pv  =  f  mnc2,  the  right-hand 
side  is  composed  of  factors  which  are  constant  at  constant  temper- 
ature, and  therefore  the  product,  pv,  must  be  constant  also  under 
similar  conditions.  This  is  clearly  Boyle's  law. 


54  THEORETICAL  CHEMISTRY 

Law  of  Gay-Lussac.    The  kinetic  equation  may  be  written  in 
the  form 

2   1 

PV  =     * 


The  kinetic  energy  of  a  single  molecule  being  represented  by 
1/2  me2,  the  total  kinetic  energy  of  the  molecules  of  the  gas  will 
be  1/2  mnc2.  Therefore  the  product  of  the  pressure  and  volume  of 
the  gas  is  equivalent  to  two-thirds  of  the  kinetic  energy  of  its  molecules. 
A  corollary  to  this  proposition  is  that  at  constant  pressure,  the 
average  kinetic  energy  of  the  molecules  in  equal  volumes  of  different 
gases  is  the  same.  The  law  of  Gay-Lussac  teaches  that  at  constant 
volume,  the  pressure  of  a  gas  is  directly  proportional  to  its  abso- 
lute temperature.  Taking  this  together  with  the  fact  that  the 
pressure  of  a  gas  at  constant  volume  is  directly  proportional  to 
the  mean  kinetic  energy  of  its  molecules,  it  follows  that  the  mean 
kinetic  energy  of  the  molecules  of  a  gas  is  directly  proportional  to  its 
absolute  temperature.  Thus  we  see  that  the  absolute  temperature  of 
a  gas  is  a  measure  of  the  mean  kinetic  energy  of  its  molecules.  This 
deduction  is  partially  based  upon  the  experimentally-determined 
law  of  Gay-Lussac.  Having  obtained  a  definiton  of  temperature 
in  terms  of  kinetic  energy,  it  is  easy  to  derive  Gay-Lussac's  law 
from  the  fundamental  kinetic  equation.  Writing  the  equation  in 
the  form 

pv  =  2'2mnc2> 

it  is  apparent  that  pv  is  directly  proportional  to  the  total  kinetic 
energy  of  the  gas  molecules,  or  in  other  words,  is  directly  propor- 
tional to  its  absolute  temperature,  which  is  the  most  general 
statement  of  Gay-Lussac  's  law. 

Law  of  Avogadro.     If  equal  volumes  of  two  different  gases  are 
measured  under  the  same  pressure,  we  will  have 

pv  =  1/3  ttimiCi2  =  1/3  nzmztf,  (1) 


where  n\  and  n^,  m\  and  ra2,  and  Ci  and  c2  denote  the  number,  mass 
and  velocity  of  the  molecules  in  the  two  gases.     If  the  gases  are 


£ 

GASES  55 

measured  at  the  same  temperature,  the  molecules  of  each  possess 
the  same  mean  kinetic  energy,  or 

l/2mid2  =  l/2w2C22.  (2) 

^Dividing  equation  (1)  by  equation  (2),  we  have 

or  under  the  same  conditions  of  temperature  and  pressure  equal 
volumes  of  the  two  gases  contain  the  same  number  of  molecules. 
This  is  the  law  of  Avogadro. 

Law  of  Graham.     If  the  fundamental  kinetic  equation  be  solved 
for  c,  we  have 

c  = 

but  v/mn  =  1/d,  where  d  is  the  density  of  the  gas,  and  therefore 
we  may  write 


=  VT- 

If  the  pressure  remains  constant  it  is  evident  that  the  mean  veloc- 
ities of  the  molecules  of  two  gases  are  inversely  proportional  to 
the  square  roots  of  their  densities,  a  law  which  was  first  enunciated 
by  Graham  in  1833  as  the  result  of  his  experiments  on  gaseous 
diffusion. 

Mean  Velocity  of  Translation  of  a  Gaseous  Molecule.  By 
substituting  appropriate  values  for  the  various  magnitudes  in  the 
equation 

c  = 

it  is  possible  to  calculate  the  mean  velocity  of  the  molecules  of 
any  gas.     Thus,  for  the  molecule  of  hydrogen  at  0°  and  76  cm. 
pressure,  p  =  76  X  13.59  =  1033.3  gr.  per  sq.  cm.  =  1033.3  X  980.6 
dynes  per  sq.  cm.,  v  =  22,400  cc.,  and  mn  =  2.016  gr. 
Substituting  these  values  in  the  above  equation  we  have, 


3  X  1033.3  X  980.6  X  22,400      100  nAA 

-  =  138,900  cm.  per  sec. 


56  THEORETICAL  CHEMISTRY 

Thus  at  0°  the  molecule  of  hydrogen  moves  with  a  speed  slightly 
greater  than  one  mile  per  second.  This  enormous  speed  is  only 
attained  along  the  mean  free  path,  the  frequent  collisions  with 
other  molecules  rendering  the  actual  speed  much  less  than  that 
calculated. 

Equation  of  Van  der  Waals.  As  has  been  pointed  out  in  a 
previous  paragraph,  the  gas  laws  are  merely  limiting  laws  and 
while  they  hold  quite  well  up  to  pressures  of  about  2  atmospheres, 
above  this  pressure  the  differences  between  the  observed  and  cal- 
culated values  become  steadily  larger.  In  the  case  of  hydrogen, 
Natterer  was  the  first  to  show  that  the  product  of  pressure  and 
volume  is  invariably  higher  than  it  should  be.  A  possible  explana- 
tion of  this  departure  from  the  gas  laws  was  offered  by  Budde, 
who  proposed  that  the  volume,  v,  in  the  equation  pv  =  RT,  should 
be  corrected  for  the  volume  occupied  by  the  molecules  them- 
selves. If  this  volume  correction  be  denoted  by  6,  then  the  gas 
equation  becomes 

p  (v  -  b)  =  RT, 

where  b  is  a  constant  for  each  gas.  Budde  calculated  the  value 
of  b  for  hydrogen  and  found  it  to  remain  constant  for  pressures 
varying  from  1000  to  2800  meters  of  mercury. 

While  Budde's  modification  of  the  gas  equation  is  quite  satis- 
factory in  the  case  of  hydrogen,  it  fails  when  applied  to  other  gases. 
In  general,  the  compressibility  at  low  pressures  is  considerably 
greater  than  can  be  accounted  for  by  Boyle's  law.  The  compressi- 
bility reaches  a  minimum  value,  and  then  increases  rapidly  so  that 
pv  passes  through  the  value  required  by  the  law.  This  suggests 
that  there  is  some  other  correction  to  be  applied  in  addition  to 
the  volume  correction  introduced  into  the  gas  equation  by  Budde. 
Van  der  Waals  pointed  out  in  1879,  that  in  the  deduction  of  Boyle's 
law  by  means  of  the  fundamental  kinetic  equation,  the  tacit 
assumption  is  made  that  the  molecules  exert  no  mutual  attraction. 
While  this  assumption  is  undoubtedly  justifiable  when  the  gas  is 
subjected  to  a  very  low  pressure,  it  no  longer  remains  so  when 
the  gas  is  strongly  compressed.  A  little  consideration  will  make 
it  apparent  that  when  increased  pressure  is  applied  to  a  gas,  the 


GASES  57 

resulting  volume  will  become  less  than  that  calculated,  owing  to 
molecular  attraction.  In  other  words  the  molecular  attraction 
and  the  applied  pressure  act  in  the  same  direction  and  the  gas 
behaves  as  if  it  were  subjected  to  a  pressure  greater  than  that 
actually  applied.  Van  der  Waals  showed  that  this  correction  is 
inversely  proportional  to  the  square  of  the  volume,  and  since  it 
augments  the  applied  pressure  the  expression  p  +  a/v2  is  sub- 
stituted for  p  in  the  gas  equation,  a  being  the  constant  of  molecular 
attraction.  The  corrected  equation  then  becomes 

(p  +  a/v2)  (v-b)  =  RT. 

This  is  known  as  the  equation  of  Van  der  Waals.  It  is  applicable 
not  only  to  strongly  compressed  gases,  but  also  to  liquids  as  well. 
While  it  will  be  given  detailed  consideration  in  a  subsequent  chapter, 
it  may  be  of  interest  to  point  out  at  this  time  the  satisfactory  ex- 
planation which  it  offers  of  the  experimental  results  of  Amagat, 
to  which  we  have  already  made  reference,  (page  49).  When  v  is 
large,  both  b  and  a/v2  become  negligible,  and  Van  der  Waals' 
equation  reduces  to  the  simple  gas  equation,  pv  =  RT.  We  may 
predict,  therefore,  that  any  influence  tending  to  increase  v  will 
cause  the  gas  to  approach  more  nearly  to  the  ideal  condition.  This 
is  in  accord  with  the  results  of  Amagat's  experiments,  which  show 
that  an  increase  of  temperature  at  constant  pressure,  or  a  diminu- 
tion of  pressure  at  constant  temperature,  causes  the  gas  to  tend  to 
follow  the  simple  gas  laws.  The  equation  also  offers  a  satisfactory 
explanation  of  the  exceptional  behavior  of  hydrogen  when  it  is 
subjected  to  pressure.  As  we  have  seen,  pv  for  all  gases,  except 
hydrogen,  diminishes  at  first  with  increasing  pressure,  reaches  a 
minimum  value,  and  then  increases  regularly.  Since  the  volume 
correction  in  Van  der  Waals'  equation  acts  in  opposition  to  the 
attraction  correction,  it  is  apparent  that  at  low  pressures  the  effect 
of  attraction  preponderates,  while  at  high  pressures  the  volume 
correction  is  relatively  of  more  importance.  At  some  intermediate 
pressure  the  two  corrections  counterbalance  each  other,  and  it  is 
at  this  point  that  the  gas  follows  Boyle's  law  strictly.  The 
exceptional  behavior  of  hydrogen  may  be  accounted  for  by  making 
the  very  plausible  assumption  that  the  attraction  correction  is 


58  THEORETICAL  CHEMISTRY 

negligible  at  all  pressures  in  comparison  with  the  volume  correc- 
tion. 

Vapor  Density  and  Molecular  Weight.  As  has  been  pointed  out 
in  an  earlier  chapter,  when  a  substance  can  be  obtained  in  the  gas- 
eous state,  the  determination  of  its  molecular  weight  resolves  itself 
into  finding  the  mass  of  that  volume  of  vapor  which  will  occupy 
22.4  liters  at  0°  and  76  cm.  It  is  inconvenient  to  weigh  a  volume 
of  gas  or  vapor  under  standard  conditions  of  temperature  and 
pressure,  but  by  means  of  the  gas  laws  the  determination  made 
at  any  temperature  and  under  any  pressure  can  be  reduced  to 
standard  conditions.  For  example,  suppose  v  cc.  of  gas  are  found 
to  weigh  w  grams  at  t°  and  p  cm.  pressure,  then  the  weight  hi  grams 
of  22.4  liters  or  22,400  cc.  at  0°  and  76  cm.  will  be  given  by  the  fol- 
lowing proportion,  in  which  M  denotes  the  molecular  weight  of  the 
substance  :  — 

PP       .  M  .  ?6  X  22,400 
*~  ~273~~ 


or 

,  ,      w  X  76  X  22,400  X  (t  +  273) 
273  pv 

The  determination  of  vapor  density  may  be  effected  in  either  of 
two  ways;  (1)  we  may  determine  the  mass  of  a  known  volume  of 
vapor  under  definite  conditions  of  temperature  and  pressure,  or 
(2)  we  may  determine  the  volume  of  a  known  mass  under  definite 
conditions  of  temperature  and  pressure.  There  are  a  variety  of 
methods  for  the  determination  of  vapor  density;  but  for  our  pur- 
pose it  will  be  necessary  to  describe  but  two  of  them.  In  the 
method  of  Regnault  the  mass  of  a  definite  volume  of  vapor  is 
determined,  while  in  the  method  due  to  Victor  Meyer  we  measure 
the  volume  of  a  known  mass. 

Method  of  Regnault.  In  this  method  which  is  especially  adapted 
to  permanent  gases,  use  is  made  of  two  spherical  glass  bulbs 
(Fig.  15)  of  approximately  the  same  capacity,  each  bulb  being 
provided  with  a  well-ground  stop-cock.  By  means  of  an  airpump 
one  bulb  in  evacuated  as  completely  as  possible,  and  is  then  filled, 
at  definite  temperature  and  pressure,  with  the  gas  whose  density 


GASES  59 

is  to  be  determined.  The  stop-cock  is  then  closed  and  the  bulb 
weighed,  the  second  bulb  being  used  as  a  counterpoise.  The  use 
of  the  second  bulb  is  largely  to  avoid  the 
troublesome  corrections  for  air  displacement 
and  for  moisture,  each  bulb  being  affected  in 
the  same  way  and  to  nearly  the  same  extent. 
The  volume  of  the  bulb  may  be  obtained  by 
weighing  it  first  evacuated,  and  then  filled  with 
distilled  water  at  known  temperature.  From 
these  results  we  may  calculate  the  mass  per  unit 
of  volume;  or  we  may  substitute  the  values  of 
w,  v,  p  and  t  in  the  above  formula  and  calcu- 
late M,  the  molecular  weight.  This  method 
was  used  by  Morley  *  in  his  epoch-making  re- 
search on  the  densities  of  hydrogen  and  oxygen. 
Method  of  Victor  Meyer.  In  the  method 
of  Victor  Meyer,  a  weighed  amount  of  the 
substance  is  vaporized,  and  the] volume  which  it  would 
occupied  at  the  temperature  of  the  room  and  under  existing 
barometric  pressure  is  determined.  The  apparatus  of  Meyer, 
shown  in  Fig.  16,  consists  of  an  inner  glass  tube  A,  about  1 
cm.  in  diameter  and  75  cm.  in  length.  This  tube  is  expanded 
into  a  bulb  at  the  lower  end,  while  at  the  top  it  is  slightly  en- 
larged and  is  furnished  with  two  side  tubes  C  and  E.  The  tube 
A  is  suspended  inside  a  heating  jacket  B,  containing  some  liquid 
the  boiling  point  of  which  is  about  20°  higher  than  the  vaporizing 
temperature  of  the  substance  whose  vapor  density  is  to  be  de- 
termined. The  side  tube  E  dips  beneath  the  surface  of  water  in  a 
pneumatic  trough  G,  and  serves  to  convey  the  air  displaced  from 
A  to  the  eudiometer  F.  By  means  of  the  side  tube  C,  and  the  glass- 
rod  D,  the  small  bulb  containing  the  substance  can  be  dropped  to 
the  bottom  of  A.  To  carry  out  a  determination  of  vapor  density 
with  this  apparatus,  the  liquid  in  B  is  heated  to  boiling  and 
the  sealed  bulb  V,  containing  a  weighed  amount  of  the  substance, 
is  placed  in  position  on  the  rod  D,  the  corks  being  tightly  inserted. 

*  Smithsonian  Contributions  to  Knowledge,  (1895). 


60 


THEORETICAL  CHEMISTRY 


When  bubbles  of  air  cease  to  issue  from  E  in  the  pneumatic  trough, 
showing  that  the  temperature  within  A  is  constant,  the  eudiometer 
F,  full  of  water,  is  placed  over  the  mouth  of  E,  and  the  bulb  V  is 
allowed  to  drop  by  drawing  aside  the  rod  D.  Air  bubbles  immedi- 
ately begin  to  issue  from  E  and  to  collect  in  the  eudiometer.  When 
the  air  ceases  to  collect,  the  eudiometer  is  closed  by  the  thumb  and 


Fig.  16. 

is  removed  to  a  large  cylinder  of  water  where  it  is  allowed  to  stand 
long  enough  to  acquire  the  temperature  of  the  room.  It  is  then 
raised  or  lowered  until  the  level  of  water  inside  and  outside  is 
the  same,  when  the  volume  of  air  is  carefully  read  off.  In  this 
method,  the  substance  on  vaporizing  displaces  an  equal  volume 
of  air  which  is  collected  and  measured,  this  observed  volume  being 


GASES  61 

that  which  the  vapor  would  occupy  after  reduction  to  the  condi- 
tions under  which  the  air  is  measured.  It  is  evident  that  in  this 
method  we  do  not  require  a  knowledge  of  the  temperature  at 
which  the  substance  vaporizes.  Since  the  air  is  measured  over 
water,  the  pressure  to  which  it  is  subjected  is  that  of  the  atmos- 
phere diminished  by  the  vapor  pressure  of  water  at  the  temperature 
of  the  experiment.  The  method  of  calculating  molecular  weights 
from  the  observations  recorded  may  be  illustrated  by  the  follow- 
ing example:  —  0.1  gram  of  benzene  (C6H6)  was  weighed  out,  and 
when  vaporized,  32  cc.  of  air  were  collected  over  water  at  17°  and 
750  mm.  pressure.  The  vapor  pressure  of  water  at  17°  is  14.4 
mm.,  and  the  actual  pressure  exerted  by  the  gas  is  750  —  14.4  = 
735.6  mm.  Substituting  in  the  proper tipn 

760  X  22,400 
=       : 

and  solving  for  M  we  have 

u  =  0.1  X  760  X  22,400  X  (17  +  273) 
273  X  735.6  X  32 

The  result  agrees  fairly  well  with  the  molecular  weight  of  benzene 
(78.05)  calculated  from  the  formula. 

Unless  a  vapor  follows  the  gas  laws  very  closely,  the  value  of  the 
molecular  weight  obtained  by  the  method  of  Victor  Meyer  will  be 
only  approximate,  but  this  approximate  value  will  be  sufficiently 
near  to  the  true  molecular  weight  to  enable  us  to  choose  between 
the  simple  formula  weight,  given  by  chemical  analysis,  and  some 
multiple  of  it. 

Results  of  Vapor-Density  Determinations.  As  the  result  of 
numerous  vapor-density  determinations  extending  over  a  wide 
range  of  temperatures,  much  important  data  has  been  collected 
concerning  the  number  of  atoms  contained  in  the  molecules  of  a 
large  number  of  chemical  compounds.  The  molecular  weights 
.of  most  of  the  elementary  gases  are  double  their  atomic  weights, 
showing  that  their  molecules  are  diatomic.  In  like  manner  the 
molecular  weights  of  mercury,  zinc,  cadmium  and,  in  fact,  all  of 
the  vaporizable  metallic  elements  have  been  found  to  be  identi- 
cal with  their  atomic  weights.  The  molecules  of  sulphur, 


62  THEORETICAL  CHEMISTRY 

arsenic,  phosphorus  and  iodine  are  polyatomic,  if  they  are  not 
heated  to  too  high  a  temperature.  The  investigations  of  Meyer 
and  others  have  shown  that  the  vapor  densities  of  a  large  number 
of  substances  diminish  as  the  temperature  is  increased.  In  other 
words  as  the  temperature  is  raised  the  number  of  atoms  contained 
in  the  molecules  decreases.  The  molecular  weight  of  sulphur,  cal- 
culated from  its  vapor  density  at  temperatures  below  500°,  corre- 
sponds to  the  formula  /S8.  If  the  vapor  of  sulphur  is  heated  to 
1100°,  the  molecular  weight  corresponds  to  the  formula  S2.  In 
fact,  sulphur  in  the  form  of  vapor  may  be  represented  by  the  formu- 
las Ss,  S^}  $2,  or  even  S  according  to  the  temperature  at  which  its 
vapor  density  is  determined.  Iodine  behaves  similarly,  the  mole- 
cules being  diatomic  between  200°  and  600°,  while  at  temperatures 
above  1400°  the  vapor  density  has  about  one-half  its  value  at  the 
lower_temperature,  showing  a  complete  breaking  down  of  the  dia- 
tomic molecules  into  single  atoms.  Heating  to  yet  higher  tem- 
peratures has  failed  to  reveal  any  further  decomposition.  This 
phenomenon  is  not  confined  to  the  molecules  of  the  elements  alone, 
but  is  also  met  with  in  the  case  of  the  molecules  of  chemical  com- 
pounds. The  vapor  density  of  arsenious  oxide  between  500°  and 
700°  corresponds  to  the  formula  As406.  As  the  temperature  is 
raised,  the  vapor  density  becomes  steadily  smaller  until,  at  1800°,  the 
calculated  molecular  weight  corresponds  to  the  formula  As2O3.  In 
like  manner  ferric  and  aluminium  chlorides  have  been  shown  to 
have  molecular  weights  at  low  temperatures  corresponding  to  the 
formulas,  Fe2Cl6  and  A12C16.  The  commonly-used  formulas,  FeCl3 
and  A1C1.3,  represent  their  molecular  weights  at  high  temperatures 
only.  The  experimental  difficulties  attending  vapor  density  de- 
terminations increase  as  the  temperature  is  raised,  owing  chiefly  to 
the  deformation  of  the  apparatus  when  the  material  of  which  it  is 
constructed  approaches  its  melting  point.  Glass  which  can  be  used 
at  relatively  low  temperatures  only,  has  been  replaced  by  specially 
resistant  varieties  of  porcelain  which  may  be  used  up  to  tempera- 
tures of  1500°  or  1600°.  Platinum  vessels  retain  their  shape  up  to 
temperatures  between  1700°  and  1800°.  Measurements  up  to 
2000°  have  recently  been  effected  by  Nernst  and  his  pupils.*  In 

*  Wartenberg.  Zeit.  anorg.  Chem.,  56,  320  (1907). 


GASES  63 

their  experiments  use  was  made  of  a  vessel  of  iridium,  the  inside 
and  outside  of  which  was  surrounded  with  a  cement  of  magnesia 
and  magnesium  chloride,  the  entire  apparatus  being  heated  electri- 
cally. With  this  apparatus  they  showed  that  the  molecular 
weight  of  sulphur  between  1800°  and  2000°  is  48,  indicating  that 
the  diatomic  molecule  is  approximately  50  per  cent  broken  down 
into  single  atoms. 

Abnormal  Vapor  Densities.  In  all  of  the  cases  cited  above 
the  molecular  weight  calculated  from  the  vapor  density  corre- 
sponds either  with  the  simple  formula  weight,  as  determined  by 
chemical  analysis,  or  with  a  multiple  thereof.  In  no  case  is  there 
any  evidence  of  a  breaking  down  of  the  simple  molecule  into  its 
constituents.  Substances  are  known,  however,  the  molecular 
weights  of  which,  calculated  from  their  vapor  densities,  are  less 
than  the  sum  of  the  atomic  weights  of  their  constituents.  For 
example,  the  vapor  density  of  ammonium  chloride  was  found  to 
be  0.89,  while  that  corresponding  to  the  formula  NH4C1  should  be 
1.89.  Similar  results  have  been  obtained  with  phosphorus  penta- 
chloride,  nitrogen  peroxide,  chloral  hydrate  and  numerous  other 
substances.  The  phenomenon  can  be  explained  in  either  of  the  two 
following  ways:  (1)  that  the  molecule  has  undergone  a  complete 
disruption,  or  (2)  that  the  substance  does  not  follow  the  law  of 
Avogadro.  Until  the  former  explanation  was  shown  to  be  correct, 
the  latter  was  accepted  and  for  a  time  the  law  of  Avogadro  fell  into 
disrepute.  In  1857,  Deville  showed  that  numerous  chemical  com- 
pounds are  broken  down  or  "  dissociated "  at  high  temperatures. 
Shortly  afterward  Kopp  suggested  that  the  abnormal  vapor 
densities  of  such  substances  as  ammonium  chloride,  phosphorus 
pentachloride,  etc.,  might  be  due  to  thermal  dissociation.  If 
ammonium  chloride  underwent  complete  dissociation,  one  molecule 
of  the  salt  would  yield  one  molecule  of  ammonia  and  one  molecule 
of  hydrochloric  acid  gas,  and  the  vapor  density  of  the  resulting 
mixture  would  be  one-half  of  that  of  the  undissociated  substance, 
a  deduction  in  complete  agreement  with  the  results  of  experiment. 
It  remained  to  prove  that  the  products  of  this  supposed  dissocia- 
tion were  actually  present. 

The  first  to  offer  an  experimental  demonstration  of  the  simul- 


64 


THEORETICAL  CHEMISTRY 


taneous  formation  of  ammonia  and  hydrochloric  acid,  when  ammon- 
ium chloride  is  heated,  was  Pebal.*  The  apparatus  which  he  de- 
vised for  this  purpose  is  shown  in  Fig.  17.  It  consisted  of  two  tubes, 
T  and  t,  the  latter  being  placed  within  the  former  as  indicated  in 
the  sketch.  Near  the  top  of  the  inner  tube,  which  was  drawn  down 
to  a  smaller  diameter,  was  a  porous  plug  of  asbestos,  (7,  upon  which 
was  placed  a  little  ammonium  chloride.  A  stream  of  dry  hydro- 


Hydrogen 


Hydrogen 


Fig.  17. 


gen  was  passed  through  the  apparatus  by  means  of  the  tubes  A  and 
Bj  the  former  entering  the  outer  tube  and  the  latter  the  inner 
tube.  The  entire  apparatus  was  heated  to  a  temperature  above 
that  necessary  to  vaporize  the  ammonium  chloride.  If  the  salt 
undergoes  dissociation  into  ammonia  and  hydrochloric  acid,  the 
former  being  less  dense  than  the  latter,  would  diffuse  more 
rapidly  through  the  plug  C  and  the  vapor  below  the  plug  would 

*  Lieb.  Ann.,  123,  199  (1862). 


GASES 


65 


be  relatively  richer  in  ammonia  than  the  vapor  above  it.  The 
current  of  hydrogen  through  B  would  therefore  sweep  out  from 
the  lower  part  of  t  an  excess  of  ammonia,  while  the  current  through 
A  would  carry  out  from  T  an  excess  of  hydrochloric  acid.  By 
holding  strips  of  moistened  litmus  paper  in  the  currents  of  gas 
issuing  from  E  and  F,  it  was  possible  for  Pebal  to  test  the  correct- 
ness of  Kopp's  idea.  He  found  that  the  gas  issuing  from  E  had 
an  acid  reaction  while  that  escaping  from  F  had  an  alkaline  reac- 
tion. It  would  at  first  sight  appear  that  Pebal  had  demonstrated 


Nitrogen 


Fig.  18. 

beyond  question  that  ammonium  chloride  undergoes  dissociation 
into  ammonia  and  hydrochloric  acid. 

It  was  pointed  out,  however,  that  Pebal  had  heated  the  ammon- 
ium chloride  in  contact  with  a  foreign  substance,  asbestos,  and 
that  this  might  have  acted  as  a  catalyst,  promoting  the  decomposi- 
tion into  ammonia  and  hydrochloric  acid.  This  objection  was 
removed  by  the  ingenious  experiment  of  Than.*  He  devised  a 
modification  of  Pebal's  apparatus,  as  shown  in  Fig.  18.  In  the 
horizontal  tube,  AB,  the  ammonium  chloride  was  placed  at  F  and  a 

*  Lieb.  Ann.,  131,  129  (1864). 


66  THEORETICAL  CHEMISTRY 

porous  plug  of  compressed  ammonium  chloride  was  introduced  at 
G.  The  tube  was  heated  and  nitrogen  passed  in  at  C.  The 
reactions  of  the  currents  of  gas  issuing  at  D  and  E  were  tested 
with  litmus  as  in  Pebal's  experiment  and  it  was  found  that  the 
gas  escaping  from  D  was  alkaline,  while  that  issuing  from  E  was 
acid.  This  experiment  proved  beyond  question  that  the  vapor 
of  ammonium  chloride  is  thermally  dissociated  into  ammonia 
and  hydrochloric  acid.  Experiments  on  other  substances  whose 
vapor  densities  are  abnormally  small  show  that  a  similar  explan- 
ation is  applicable,  and  thus  furnish  a  confirmation  of  the  law  of 
Avogadro. 

Calculation  of  the  Degree  of  Dissociation.  Since  the  density 
of  a  dissociating  vapor  decreases  with  increase  in  temperature, 
it  is  important  to  be  able  to  calculate  the  degree  of  dissocation  at 
any  one  temperature.  This  is  clearly  equivalent  to  ascertaining 
the  extent  to  which  the  reaction 

NH4Ci<=±NH8+-HCl 

has  proceeded  from  left  to  right.  This  can  be  determined  easily 
from  the  relation  of  vapor  density  to  dissociation.  If  we  start 
with  one  molecule  of  gas  and  let  a  represent  the  percentage  dis- 
sociation, then  1  —  a  will  denote  the  percentage  remaining  un- 
dissociated.  If  one  molecule  of  gas  yields  n  molecules  of  gaseous 
products,  the  total  number  of  molecules  present  at  any  time  will 
be 

(1  -  a)  +  na  =  1  +  (n  -  1)  a. 

The  ratio  1  :  1  +  (n  —  1)  a  will  be  the  same  as  the  ratio  of  the 
density  dz  of  the  dissociated  gas  to  its  density  in  the  undissociated 
state  di,  or 

1  :1  +  (n-  I)  a  =  d2  id^ 

solving  this  proportion  for  a,  we  have 


The  vapor  density  of  nitrogen  peroxide  has  been  measured  by  E. 
and  L.  Natanson,*  and  the  degree  of  dissociation  at  the  different 

*  Wied.  Ann.,  24,  454  (1885);  27,  606  (1886). 


GASES 


67 


temperatures  calculated  by  means  of  the  preceding  formula.     The 
following  table  gives  their  results. 

The  course  of  the  dissociation  is  shown  in  the  accompanying 
illustration,  Fig.  19,  in  which  the  abscissae  represent  temperature 
and  the  ordinates,  percentage  dissociation.  It  will  be  observed 


100 


90  - 


80 


gzo- 

' 


f50 

$40 
30 
20 
10 


20 


40 


80  100 

Temperature 


140 


160 


Fig.  19. 


that  the  dissociation  of  nitrogen  peroxide  is  at  first  nearly  pro- 
portional to  the  temperature.  It  then  increases  more  rapidly 
until,  when  about  four-fifths  of  the  molecules  of  ^64  are  broken 
down,  the  dissociation  proceeds  slowly  to  completion. 

Specific  Heat.  The  addition  of  heat  energy  to  a  body  causes 
its  temperature  to  rise.  The  ratio  of  the  amount  of  heat  supplied 
to  the  resulting  rise  in  temperature  is  called  the  heat  capacity  of 
the  body;  obviously  its  value  is  dependent  upon  the  initial  temper- 


THEORETICAL  CHEMISTRY 


DISSOCIATION  OF  NITROGEN  PEROXIDE,   N2O4. 

ATMOSPHERIC  PRESSURE. 
(Density  of  N*04=3.18;    of  N02+NO2  =  1.59;  of  air  =  1.00) 


Temperature, 
(degrees) 

Density  of  Gas. 

Percentage  Dis- 
sociation. 

26:7 

2.65 

19.96 

35.4 

2.53 

25.65 

39.8 

2.46 

29.23 

49.6 

2.27 

40.04 

60.2 

2.08 

52.84 

70.0 

1.92 

65.57 

80.6 

1.80 

76.61 

90.0 

1.72 

84.83 

100.1 

1.68 

89.23 

111.3 

1.65 

92.67 

121.5 

1.62 

96.23 

135.0 

1.60 

98.69 

154.0 

1.58 

100.00 

ature  of  the  body.  The  specific  heat  of  a  substance  may  be  defined 
as  the  heat  capacity  of  unit  mass  of  the  substance.  If  dt  represents 
the  increment  of  temperature  due  to  the  addition  of  dQ  units  of 
heat  energy  to  m  grams  of  any  substance,  then  its  specific  heat,  c, 
will  be  given  by  the  equation 

-1M. 

~m    dt 

Specific  Heat  at  Constant  Pressure  and  Constant  Volume.  It 
is  well  known  that  the  specific  heat  of  a  gas  depends  upon  the 
conditions  under  which  it  is  determined.  If  a  definite  mass  of 
gas  is  heated  under  constant  pressure,  the  value  of  the  specific 
heat,  cp,  is  different  from  the  value  of  the  specific  heat,  cv,  ob- 
tained when  the  pressure  varies  and  the  volume  remains  con- 
stant. The  value  of  cp  is  invariably  greater  than  that  of  cv. 
When  heat  is  supplied  to  a  gas  at  constant  pressure  not  only  does 
its  temperature  rise,  but  it  also  expands,  and  thus  does  external 
work.  On  the  other  hand,  if  the  gas  be  heated  in  such  a  way  that 
its  volume  cannot  change,  none  of  the  heat  supplied  will  be  used 
in  doing  external  work,  and  consequently  its  heat  capacity  will 


GASES  69 

be  less  than  when  it  is  heated  under  constant  pressure.  The 
recognition  by  Mayer  in  1841  of  the  cause  of  this  difference  between 
the  two  specific  heats  of  a  gas  led  him  to  his  celebrated  calculation 
of  the  mechanical  equivalent  of  heat,  and  the  enunciation  of  the 
first  law  of  thermodynamics.  Mayer  observed  that  the  differ- 
ence between  the  quantity  of  heat  necessary  to  raise  the  temper- 
ature of  1  gram  of  air  1°  C.  at  constant  pressure,  and  at  constant 
volume  respectively,  was  0.0692  calorie,  or 

Cp  -  Cv  =  0.0692  cal. 

That  is  to  say,  0.0692  calorie  is  the  amount  of  heat  energy  which 
is  equivalent  to  the  work  required  to  expand  1  gram  of  air  1/273 
of  its  volume  at  0°.  Imagine  1  gram  of  air  at  0°  enclosed  within 
a  cylinder  having  a  cross-section  of  one  square  centimeter,  and 
furnished  with  a  movable,  frictionless  piston.  Since  1  gram  of 
air  under  standard  conditions  of  temperature  and  pressure  occu- 
pies 773.3  cc.,  the  distance  between  the  piston  arid  the  bottom  of 
the  cylinder  will  be  773.3  cm.  If  the  temperature  be  raised  from 
0°  to  1°,  the  piston  will  rise  1/273  X  773.3  =  2.83  cm.,  and  since 
the  pressure  of  the  atmosphere  is  1033.3  grams  per  square  centi- 
meter, the  external  work  done  by  the  expanding  gas  will  be 

1033.3  X  2.83  =  2923.4  gr.  cm. 

This  is  evidently  equivalent  to  0.0692  calorie  and  therefore,  the 
equivalent  of  1  calorie  in  mechanical  units,  J,  will  be 


J  =       =  42'245>  gr- 


a  value  agreeing  very  well  with  the  best  recent  determinations  of 
the  mechanical  equivalent  of  heat. 

The  difference  between  the  two  specific  heats  may  be  easily 
calculated  in  calories  from  the  fundamental  gas  equation.  Start- 
ing with  1  mol.  of  gas,  and  remembering  that  when  a  gas  expands 
at  constant  pressure,  the  product  of  pressure  and  change  in  volume 
is  a  measure  of  the  work  done,  we  have,  at  temperature  TI°, 

pvi  =  RTl} 
where  vi  is  the  molecular  volume.     Raising  the  temperature  to 


70  THEORETICAL  CHEMISTRY 

T2°,  the  corresponding  molecular  volume  being  vz,  we  have  for 
the  work  done  during  expansion 

P  fa  -vi)=R  (T2  -  Ti). 
If  T2  —  Ti  =  1°,  then  the  equation  reduces  to 

P  (vz  -  vi)  =  R. 

Since  the  difference  between  the  molecular  heats  *  at  constant 
pressure  and  constant  volume  is  equivalent  to  the  external  work 
involved  when  the  temperature  of  1  mol.  of  gas  is  raised  1°,  we  have 

M  (cp  -  cv)  =  p  (vz  -  vi), 

where  M  is  the  molecular  weight  of  the  gas;  and  therefore 
M  (cp  —  cv)  =  R  =  2  calories. 

In  words,  the  difference  of  the  molecular  heats  of  any  gas  at 
constant  pressure  and  at  constant  volume  is  2  calories.  The 
specific  heat  of  a  gas  at  constant  pressure  can  be  readily  deter- 
mined, by  passing  a  definite  volume  of  the  gas,  heated  under  con- 
stant pressure  to  a  known  temperature,  through  the  worm  of  a 
calorimeter  at  such  a  rate  that  a  constant  difference  is  maintained 
between  the  temperature  of  the  entering  and  the  temperature  of 
the  escaping  gas.  Thus  the  number  of  calories  which  causes  a 
definite  thermal  change  in  a  certain  volume  of  the  gas  is  deter- 
mined, and  from  this  it  is  an  easy  matter  to  calculate  the  specific 
heat,  cp.  The  molecular  heat  at  constant  pressure  for  all  gases 
approaches  the  limiting  value,  6.5,  at  the  absolute  zero.  This 
relation,  due  to  Le  Chatelier,  may  be  expressed  thus, 
Mcp  =  6.5  +  aT, 

where  a  is  a  constant  for  each  gas.  The  value  of  a  for  hydrogen, 
oxygen,  nitrogen  and  carbon  monoxide  is  0.001,  for  ammonia, 
0.0071  and  for  carbon  dioxide,  0.0084.  As  the  complexity  of  the 
gas  increases  the  value  of  a  becomes  numerically  greater. 

The  experimental  determination  of  the  specific  heat  of  a  gas  at 
constant  volume  is  difficult  and  the  results  obtained  are  not 
trustworthy.  The  chief  cause  of  the  inaccuracy  of  the  results 

*  The  molecular  heat  of  a  gas  is  equal  to  the  product  of  its  specific  heat  and  its 
molecular  weight. 


GASES 


71 


is  that  the  vessel  containing  the  gas  absorbs  so  much  more  heat 
than  the  gas  itself  that  the  correction  is  many  times  larger  than 
the  quantity  to  be  measured.  The  specific  heat  at  constant  vol- 
ume is  almost  always  obtained  by  indirect  methods,  as  for  example 
by  means  of  the  preceding  formula 

M  (cp  -  cv)  =  R  =  2  cal, 

in  which  the  values  of  M  and  cp  are  known. 

The  molecular  heats  of  some  of  the  commoner  gases  and  vapors 
are  given  in  the  subjoined  table  together  with  the  ratio   cp/cv. 

MOLECULAR  SPECIFIC  HEATS. 


Gas. 

Mcp 

Mcv 

cp/cv=y 

Argon                    

1  66 

Helium                   

1.66 

Mercury          

.66 

Hydrogen  

6.88 

4.88 

.41 

Oxygen 

6.96 

4.96 

.40 

Nitrogen  

6.93 

4.93 

.41 

Chlorine 

8  58 

6  58 

30 

Bromine  

8.88 

6.88 

.29 

Nitric  oxide 

6  95 

4  95 

40 

Carbon  monoxide 

6  86 

4  86 

41 

Hydrochloric  acid  

6.68 

4.68 

.43 

Carbon  dioxide 

9  55 

7  55 

26 

Nitrous  oxide 

9.94 

7  94 

25 

Water 

8  65 

6  65 

28 

Sulphur  dioxide 

9.88 

7  88 

25 

Ozone                                          ...    . 

29 

Ether                                 .  . 

35  51 

33  51 

.06 

The  Ratio  of  the  Two  Specific  Heats.  There  are  two  methods 
by  which  the  ratio  cp/cv  can  be  determined  directly,  one  due  to 
Clement  and  Desormes  *  and  the  other  due  to  Kundt.f 

Method  of  Clement  and  Desormes.  The  apparatus  devised  by 
these  investigators  consists,  as  is  shown  in  Fig.  20,  of  a  glass 
balloon  flask,  A,  of  about  20  liters  capacity,  furnished  with  two 
stop-cocks,  D  and  E,  and  a  manometer,  C.  The  stop-cock  D 
has  an  aperture  nearly  as  large  as  the  diameter  of  the  neck  of  the 

*  Jour,  de  phys.,  89,  321,  428  (1819). 

f  Pogg.  Ann.,  127,  497  (1866);  135,  337,  527  (1868). 


72 


THEORETICAL  CHEMISTRY 


flask,  B.  To  determine  the  ratio  of  the  two  specific  heats,  the 
flask  is  filled  with  the  gas  under  a  pressure  slightly  greater  than 
barometric  pressure.  The  manometer  C  serves  to  measure  the 
pressure  of  the  gas  within  A.  After  the  value  of  the  pressure 
has  been  read  on  the  manometer,  the  stop-cock  D  is  opened 
momentarily  to  the  air,  thus  permitting  the  pressure  of  the  gas 
to  fall  adiabatically  to  that  of  the  atmosphere.  The  stop-cock 


Fig.  20. 

is  then  closed  and  the  flask  is  allowed  to  stand  for  a  few  moments 
until  its  contents,  which  has  cooled  by  adiabatic  expansion,  has 
regained  the  temperature  of  the  room.  The  pressure  on  the 
manometer  is  then  observed.  Let  the  initial  pressure  of  the  gas 
be  denoted  by  po,  and  atmospheric  pressure  by  P.  If  the  initial 
and  final  specific  volumes  are  denoted  by  VQ  and  v\,  then  for  an 
adiabatic  process,  we  have 

- 

Po 


GASES  73 

The  value  of  the  final  specific  volume  is  determined  from  the 
final  pressure,  pi,  by  an  application  of  Boyle's  law,  the  pressure  pi 
being  developed  isothermally. 
Thus, 

VQ^PI 
Vi    Po 

and  consequently 


Po 
or 

=  log  P  -  log  po  ^ 
log  pi  —  log  po 

Method  of  Kundt.     According  to  the  formula  of  Laplace  for  the 
velocity  of  transmission  of  a  sound  wave  in  a  gas,  we  have 


in  which  p  and  d  denote  the  pressure  and  density  of  the  gas,  and 
7  is  the  ratio  of  the  two  specific  heats.  If  the  wave  velocities  in 
two  different  gases,  whose  densities  are  d\  and  dz  under  the  same 
conditions  of  temperature  and  pressure,  be  denoted  by  v\  and  v^, 
we  may  write 


or  replacing  the  densities  of  these  gases  by  their  respective  molec- 
ular weights,  MI  and  M2,  we  have 


The  ratio  of  the  velocities  of  the  two  waves  can  be  measured  by 
means  of  the  apparatus  shown  in  Fig.  21.  A  wide  glass  tube 
about  1|  meters  in  length  is  furnished  with  two  side  tubes,  E  and 
F.  Into  one  end  of  the  tube  is  inserted  the  glass  rod  BD  which 
is  clamped  at  its  middle  point  by  a  tightly  fitting  cork,  C.  The 
other  end  of  the  tube  is  closed  by  means  of  the  plunger  A.  A 
small  amount  of  lycopodium  powder  is  placed  upon  the  bottom  of 


74  THEORETICAL  CHEMISTRY 

the  tube  and  is  distributed  uniformly  by  gently  tapping  the  walls 
of  the  tube.  The  gas  in  which  the  velocity  of  the  sound  wave 
is  to  be  determined  is  introduced  into  the  tube  through  E,  and 
the  displaced  air  escapes  at  F.  When  the  tube  is  filled,  E  and  F 
are  closed  by  means  of  rubber  caps,  and  a  piece  of  moistened 
chamois  leather  is  drawn  along  BD  causing  it  to  vibrate  longitudi- 
nally and  to  emit  a  shrill  note.  The  vibrations  are  taken  up  by  the 


Fig.  21. 

gas  in  the  tube  and  the  powder  arranges  itself  in  a  series  of  heaps 
corresponding  to  the  nodes  of  vibration.  If  the  nodes  are  not 
sharply  defined,  then  A  should  be  moved  in  or  out  until  they 
become  so.  If  Xi  is  the  distance  between  two  heaps  or  nodes, 
then  2  Xi  will  be  the  wave  length  of  the  note  emitted  by  the  rod 
BD,  and  if  n  represents  the  number  of  vibrations  per  second  of 
the  note  emitted,  we  have  for  the  velocity  of  sound  in  the  gas 


Similarly  if  a  second  gas  be  introduced  into  the  tube  we  shall 
have 


Therefore, 

5  =£.  (2) 

Vz        X2 

Substituting  in  equation  (1),  we  have 

Xi 

~  — 

X2 
or 

Xi2  Mi  ,,. 

Ti=T2v'Mr 

If  the  second  gas  is  air,  as  is  usually  the  case,  72  =  1.405  and  Mz  = 
28.74,  (mol.  wt.  of  hydrogen  •*•  density  of  hydrogen  referred  to  air, 
or  2  -f-  0.0696  =  28.74)  or  equation  (3)  becomes 


GASES  75 


Xi2 
71  =  1.405  :-=• 


X22   28.74 

Thus,  7  for  any  gas  can  be  determined  by  this  method  provided 
we  know  its  value  for  another  gas  of  known  molecular  weight. 

Specific  Heat  of  Gases  and  the  Kinetic  Theory.  In  terms  of 
the  kinetic  theory,  the  energy  of  a  gas  may  be  considered  to  be 
made  up  of  three  parts:  (1)  the  translational  energy  of  the  mole- 
cules, commonly  termed  their  kinetic  energy,  (2)  the  intramolec- 
ular kinetic  energy,  and  (3)  the  potential  energy  due  to  inter- 
atomic attraction  within  the  molecules.  When  a  gas  is  heated 
at  constant  volume  all  three  of  these  factors  of  the  total  energy 
of  the  molecule  may  be  affected.  It  is  fair  to  assume,  however, 
that  when  a  monatomic  gas,  such  as  mercury  vapor,  is  heated, 
all  of  the  heat  energy  supplied  is  used  to  augment  the  translational 
kinetic  energy  of  the  molecules.  As  we  have  seen,  the  fundamental 
kinetic  equation 

pv  =  1/3  nmc2 
may  be  written 

pv  =  2/3  •  1/2  nmc2, 

and  since  1/2  nmc2  represents  the  total  kinetic  energy  of  the  gas, 
we  have 

pv  =  2/3  kinetic  energy  of  1  mol, 
or 

kinetic  energy  of  1  mol.  =  3/2  pv. 

But  pv  =  2  T  calories,  therefore 

kinetic  energy  of  1  mol.  =  3  T  cal. 

The  kinetic  energy  of  a  constant  volume  of  any  gas  at  the  temper- 
atures TI  and  TZ)  is  given  by  the  following  equations :  — 

3/2  Plt;  =  3  Ti,  (1) 

and 

3/2p,i;  =  3!rs.  (2) 

Subtracting  (1)  from  (2)  we  obtain 

3/2  (p,  -  pO  v  =  3  (Tt  -  7U  (3) 

and  for  an  increase  in  temperature  of  1°,  (3)  becomes 
3/2  (p2  -  pi)  v  =  3  cal. 


76  THEORETICAL  CHEMISTRY 

The  molecular  kinetic  energy  of  a  monatomic  gas  at  constant 
volume  is  thus  increased  by  3  calories  for  each  degree  rise  in  tem- 
perature. As  has  already  been  shown, 

M  (cp  —  cv)  =  2  cal., 
therefore,  since  Mcv  =  3  calories,  Mcp  =  3  +  2  =  5  calories,  and 

Mcp      5 

7=-W^  =  Q  =  L66- 
Mcv      3 

This  value  of  7  is  in  perfect  agreement  with  the  results  of  the 
experiments  on  mercury  vapor  which  is  known  to  be  monatomic. 
The  converse  of  this  method  has  been  employed  by  Ramsay  to 
prove  that  the  rare  gases  of  the  atmosphere  are  monatomic,  the 
value  of  7  for  all  of  these  gases  being  1.66.  In  the  case  of  poly- 
atomic molecules  the  heat  energy  supplied  is  not  only  used  in 
increasing  their  translational  kinetic  energy,  but  also  in  the 
performance  of  work  within  the  molecule.  The  value  of  the 
internal  work  is  indeterminate,  but  it  is  without  doubt  constant 
for  any  one  gas.  If  the  internal  work  be  represented  by  a,  then 
the  value  of  the  ratio  of  the  two  specific  heats  will  be 

Mcp      5  +  a 

7=  -rjr2  =  ^— —  <  1.66  >  1 . 
M  cv      3  +  a 

Reference  to  the  table  on  p.  71,  giving  the  value  of  7  for  differ- 
ent gases,  will  show  that  this  deduction  from  the  kinetic  theory 
is  in  perfect  agreement  with  the  experimental  facts.  With 
increasing  complexity  of  the  molecule,  it  is  apparent  that  the 
amount  of  heat  expended  in  doing  internal  work  should  increase, 
and  therefore  the  specific  heat  should  increase  also.  Inspection 
of  the  table  confirms  this  deduction.  The  specific  heat  of  mona- 
tomic gases  is  independent  of  the  temperature  while  the  specific 
heat  of  polyatomic  gases  increases  slightly.  These  results  may 
justly  be  regarded  as  among  the  greatest  triumphs  of  the  kinetic 
theory  of  gases. 


GASES  77 


PROBLEMS. 

1.  The  volume  of  a  quantity  of  gas  is  measured  when  the  barometer 
stands  at  72  cm.,  and  is  found  to  be  646  cc. :  what  would  its  volume  be 
at  normal  pressure?  Ans.   612  cc. 

2.  At  what  pressure  would  the  gas  in  the  preceding  problem  have  a 
volume  of  580  cc.?  Ans.  80.19  cm. 

3.  A  certain  quantity  of  oxygen  occupies  a  volume  of  300  cc.  at  0°: 
find  its  volume  at  91°.  Ans.  400  cc. 

4.  The  weight  of  a  liter  of  air  under  standard  conditions  is  1.293  grams: 
to  what  temperature  must  the  air  be  heated  so  that  it  may  weigh  exactly 
1  gram  per  liter?  Ans.  79°.99. 

5.  At  what  temperature  will  the  volume  of  a  given  mass  of  gas  be 
exactly  double  what  it  is  at  17°?  Ans.   307°. 

6.  On  heating  a  certain  quantity  of  mercuric  oxide  it  is  found  to  give 
off  380  cc.  of  oxygen,  the  temperature  being  23°  and  the  barometric 
height  74  cm.;    what  would  be  the  volume  of  the  gas  under  standard 
conditions?  ^Ans.  341.25  cc. 

7.  A  liter  of  air  weighs  1.293  grams  under  standard  conditions.   £t 
what  temperature  will  a  liter  of  air  weigh  1  gram,  the  pressure  being  72  cm.? 

Ans.  61°.43. 

8.  A  quantity  of  air  at  atmospheric  pressure  and  at  a  temperature  of 
7°  is  compressed  until  its  volume  is  reduced  to  one-seventh,  the  temper- 
ature rising  20°  during  the  process:   find  the  pressure  at  the  end  of  the 
operation.  Ans.   7.5  atmos. 

9.  The  weight  of  a  liter  of  nitrogen  under  standard  conditions  is  1.2579 
grams.     Calculate  the  specific  gas  constant,  r.          Ans.  3007  gr.  cm. 

10.  The  time  of  outflow  of  a  gas  is  21.4  minutes,  the  corresponding 
time  for  hydrogen  is  5.6  minutes.     Find  the  molecular  weight  of  the  gas. 

.  .-  Ans.   29.2. 

11.  Calculate  the  molecular  weight  of  chloroform  from  the  following 
data:  — 

Weight  of  chloroform  taken   0 . 220  gr. 

Volume  of  air  collected  over  water   45 .0  cc. 

Temperature  of  air 20° 

Barometric  pressure  755 .0  mm. 

Pressure  of  aqueous  vapor  at  20° 17 .4  mm. 

Ans.   117. 


78  THEORETICAL  CHEMISTRY 

12.  The  density  of  a  gas  is  0.23  referred  to  mercury  vapor.    What  is 
its  molecular  weight?  Ans.  46. 

13.  Phosphorus  pentachloride  dissociates  according  to  the  equation 


The  molecular  weight  of  PC15  is  208.28.    At  182°  the  density  is  73.5  and 
at  230°  it  is  62.     Find  the  degree  of  dissociation  at  the  two  temperatures. 

Ans.   am°  =  0.417,  <xm°  =  0.68. 

14.  The  specific  heat  at  constant  volume  for  argon  is  0.075,  and  its 
molecular  weight  is  40.    How  many  atoms  are  there  in  the  molecule? 

Ans.   1. 

15.  What  is  the  specific  heat  of  carbon  dioxide  at  constant  volume,  its 
molecular  weight  being  44  and  the  temperature  50°.  Ans.   0.164. 

16.  The  specific  heat  for  constant  pressure  of  benzene  is  0.295  :  what  is 
the  specific  heat  for  constant  volume?  Ans.  0.27. 


CHAPTER  V. 
LIQUIDS. 

General  Characteristics  of  Liquids.  The  most  marked  char- 
acteristic of  the  liquid  state  is  that  a  given  mass  of  liquid  has  a 
definite  volume  but  no  definite  form.  The  volume  of  a  liquid  is 
dependent  upon  temperature  and  pressure  but  to  a  much  smaller 
degree  than  is  the  volume  of  a  gas.  The  formulas  in  which  the  vol- 
ume of  a  liquid  is  expressed  as  a  function  of  temperature  and  pres- 
sure are  largely  empirical,  and  contain  constants  dependent  upon 
the  nature  of  the  liquid.  This  is  undoubtedly  due  to  the  fact  that 
in  the  liquid  state  the  molecules  are  much  less  mobile  than  in  the 
gaseous  state.  The  distance  between  contiguous  molecules  being 
much  less  in  liquids  than  in  gases,  the  mutual  attraction  is  increased 
while  the  mobility  is  correspondingly  diminished.  That  liquids 
Represent  a  more  condensed  form  of  matter  than  gases  is  shown 
by  the  change  in  volume  which  results  when  a  liquid  is  vaporized : 
thus,  1  cc.  of  water  at  the  boiling  point  when  vaporized  at  the  same 
temperature  occupies  a  volume  of  about  1700  cc.  A  liquid  con- 
tains less  energy  than  a  gas,  since  energy  is  always  required  to 
transform  it  into  the  gaseous  state.  Since  gases  can  be  liquefied 
by  increasing  the  pressure  and  lowering  the  temperature,  and  since 
liquids  can  be  vaporized  by  lowering  the  pressure  and  increasing 
the  temperature,  it  is  apparent  that  there  is  no  generic  difference 
between  the  two  states  of  matter. 

Connection  Between  the  Gaseous  and  Liquid  States.  If  a  gas 
is  compressed  isothermally,  its  state  may  change  in  either  of  two 
ways  depending  upon  the  temperature: — (1)  The  volume  at 
first  diminishes  more  rapidly  than  the  pressure,  then  in  the  same 
ratio  and  lastly  more  slowly.  When  the  pressure  attains  a  very 
high  value  the  volume  is  but  slightly  altered.  This  case  has 
already  been  considered  in  the  preceding  chapter.  (2)  The 
volume  changes  more  rapidly  than  the  pressure  until,  when  a  cer- 

79 


80  THEORETICAL  CHEMISTRY 

tain  pressure  is  reached,  the  gas  ceases  to  be  homogeneous,  partial 
liquefaction  resulting.  For  a  constant  temperature,  the  pressure 
at  which  liquefaction  occurs  is  invariable,  while  the  volume 
steadily  diminishes  until  liquefaction  is  complete.  Only  when 
the  whole  mass  of  gas  has  been  liquefied  is  it  possible  to  increase 
the  pressure  and  then,  owing  to  the  small  compressibility  of  liquids, 
a  large  increase  in  pressure  is  required  to  produce  a  slight  dimin- 
ution in  volume.  If  the  temperature  is  above  a  certain  point, 
dependent  upon  the  nature  of  the  gas,  the  phenomena  of  com- 
pression will  follow  (1);  if  below  this  point,  the  process  will  follow 
(2).  That  a  gas  may  behave  in  either  of  the  above  ways  was 
first  clearly  recognized  by  Andrews  *  in  1869,  in  connection  with 
his  experiments  on  the  liquefaction  of  carbon  dioxide.  He  found 
that  if  carbon  dioxide  was  compressed,  keeping  the  temperature 
at  0°,  the  volume  changes  more  rapidly  than  the  pressure,  lique- 
faction resulting  when  a  pressure  of  35.4  atmospheres  was  reached. 
As  the  temperature  was  raised,  he  found  that  a  higher  pressure 
was  required  to  liquefy  the  gas,  until  at  temperatures  above 
30°.92  it  was  no  longer  possible  to  condense  the  gas  to  the  liquid 
state.  The  temperature  above  which  it  was  no  longer  possible 
to  liquefy  the  gas  he  termed  the  critical  temperature.  In  like 
manner  the  pressure  required  to  liquefy  the  gas  at  the  critical 
temperature,  he  termed  the  critical  pressure,  and  the  volume 
occupied  by  the  gas  or  the  liquid  under  these  conditions  he  called 
the  critical  volume. 

Isothermals  of  Carbon  Dioxide.  The  results  of  Andrew's 
experiments  f  on  the  liquefaction  of  carbon  dioxide  are  shown  in 
Fig.  22,  in  which  the  ordinates  represent  pressures  and  the  abscissae 
the  corresponding  volumes  at  constant  temperature.  The  curves 
obtained  by  plotting  volumes  against  pressures  at  constant 
temperatures  are  called  isothermals.  For  a  gas  which  follows 
Boyle's  law,  the  isothermals  will  be  a  series  of  equilateral  hy- 
perbolas. This  condition  is  approximately  fulfilled  by  air,  for 
which  three  isothermals  are  given  in  the  diagram.  At  48°.  1  the 
isothermal  for  carbon  dioxide  is  nearly  hyperbolic,  but  as  the 

*  Trans.  Roy.  Soc.  159,  583  (1869). 
f  loc.  cit. 


LIQUIDS 


81 


temperature  becomes  lower,  the  isothermals  deviate  more  and  more 
from  those  for  an  ideal  gas.     At  the  critical  temperature,  30°. 92, 


Carbon  Dioxide  — 


Air 


Volume 
Fig.  22. 


the  curve  is  almost  horizontal  for  a  short  distance,  showing  that  for 
a  very  slight  change  in  pressure  there  is  an  enormous  shrinkage 
in  volume.  At  still  lower  temperatures,  21°.l  and  13°.  1,  the 


82  THEORETICAL  CHEMISTRY 

horizontal  portions  of  the  curves  are  much  more  pronounced, 
indicating  that  during  liquefaction  there  is  no  change  in  pressure. 
When  liquefaction  is  complete  the  curves  rise  abruptly,  showing 
that  the  change  in  volume  is  extremely  small  for  a  large  increase 
in  pressure;  in  other  words  the  liquefied  gas  possesses  a  small 
coefficient  of  compressibility.  At  any  point  within  the  parabolic 
area,  indicated  by  the  dotted  line  ABC,  both  vapor  and  liquid  are 
coexistent  ;  at  any  point  outside,  only  one  form  of  matter,  either 
liquid  or  vapor,  is  present.  Andrew's  experiments  show  that 
there  is  no  fundamental  difference  between  a  gas  and  a  liquid. 
It  is  apparent  from  the  diagram  that  when  carbon  dioxide  is  sub- 
jected to  great  pressures  above  its  critical  temperature  it  behaves 
more  like  a  liquid  than  a  gas,  in  fact  it  is  difficult  to  determine 
whether  a  highly  compressed  gas  above  its  critical  temperature 
should  be  classified  as  a  gas  or  as  a  liquid. 

Van  der  Waals'  Equation  and  the  Continuity  of  the  Gaseous 
and  Liquid  States.  In  the  preceding  chapter  we  have  learned 
that  the  fundamental  gas  equation 

pv  =  RT 

is  only  strictly  applicable  to  an  ideal  gas,  and  that  the]  behavior 
of  actual  gases  is  represented  with  considerable  accuracy,  even  at 
high  pressures,  by  the  equation  of  Van  der  Waals, 


P       .»-     = 

If  this  equation  be  arranged  in  descending  powers  of  v,  we  have 

./,    .  RT\  .      a      ab  m 

v3  —  v2[b-i  --  +  v  ---  =  0.  (1) 

\         P  I       P       P 

This  being  a  cubic  equation  has  three  possible  solutions,  each  val- 
ue of  p  affording  three  corresponding  values  of  v;  a,  b,  R  and  T  being 
treated  as  constants.  The  three  roots  of  this  equation  are  either 
all  real,  or  one  is  real  and  two  are  imaginary,  depending  upon  the 
values  of  the  constants.  That  is  to  say,  at  one  temperature 
and  pressure  the  values  of  a  and  6  may  be  such,  that  v  has  three 
real  values,  while  at  another  temperature  and  pressure,  v  may 
have  one  real  and  two  imaginary  values.  In  the  accompanying 


LIQUIDS 


83 


diagram,  Fig.  23,  a  series  of  graphs  of  the  equation  for  different 
values  of  T  is  given.  It  will  be  observed  that  these  curves  bear 
a  striking  resemblance  to  the  isotherms  of  carbon  dioxide  estab- 
lished by  the  experiments  of  Andrews.  In  the  case  of  the  theo- 


Volume 
Fig.  23. 

retical  curves  there  are  no  sudden  breaks  such  as  appear  in  the 
actual  discontinuous  passage  from  the  gaseous  to  the  liquid  state. 
Instead  of  passing  from  B  to  D  along  the  wavelike  path  BaCbD, 
experiment  has  shown  that  the  substance  passes  directly  from  the 
state  B  to  the  state  D  along  the  straight  line  BD.  It  is  here 


84  THEORETICAL  CHEMISTRY 

that  Van  der  Waals'  equation  fails  to  apply.  As  has  been  pointed 
out  the  substance  between  these  two  points  is  not  homogeneous, 
being  partly  gaseous  and  partly  liquid.  Attempts  have  been 
made  to  realize  the  portion  of  the  curve  BaCbD  experimentally. 
By  studying  supersaturated  vapors  and  superheated  liquids  it 
has  been  found  possible  to  follow  the  theoretical  curve  for  short 
distances  between  B  and  D  without  discontinuity,  but  owing  to 
the  instability  of  the  substance  in  this  region,  it  is  evident  that 
the  complete  isothermal  and  continuous  transformation  of  a  gas 
into  a  liquid  can  never  be  effected.  Van  der  Waals  has  called 
attention  to  the  fact  that  in  the  surface  layer  of  a  liquid,  where 
unique  conditions  prevail,  it  is  quite  possible  that  such  unstable 
states  may  exist,  and  that  there  the  transition  from  liquid  to  gas 
may  in  reality  be  a  continuous  process.  The  diagram  shows  that 
as  T  increases,  the  wave-like  portion  of  the  isothermals  becomes 
less  pronounced  and  eventually  disappears,  when  the  points  B} 
C  and  D  coalesce.  At  this  point  the  three  roots  of  the  equa- 
tion become  equal,  the  volume  of  the  liquid  becoming  identical 
with  the  volume  of  the  gas.  The  substance  at  this  point  is  in  the 
critical  condition.  Since  under  these  conditions  the  three  roots 
of  the  equation 

vz  —  ( b  +  — 1  v2  +  -  v  —  —  =  0  (1) 

\          p  I          p         p 

are  equal,  we  may  write  Vi  =  v2  =  vs=  vc,  the  subscript  c  indicat- 
ing the  critical  state.     Then  equation  (1)  must  be  equivalent  to 
(v  -  Vc)*  =  v3  -  3  vcv*  +  3  vc2v  -  vcs  =  0.  (2) 

Equating  the  corresponding  coefficients  of  equations  (1)  and  (2), 
we  have 


(3) 

PC 

3».«--  (4) 

PC 

and  vc*  =  —  •  (5) 

PC 

Dividing  equation  (5)  by  equation  (4),  we  have 

vc  =  Zb,  (6) 


LIQUIDS  85 

and  substituting  this  value  in  equation  (4),  we  obtain 


Lastly,  substituting  the  values  of  vc  and  pc)  given  in  equations 
(6)  and  (7),  in  equation  (3),  we  have 


Therefore, 


Or  expressing  the  constants  a,  b  and  R  in  terms  of  the  critical 
values  of  pressure,  temperature  and  volume,  we  have 

a  =  3  pcvc*,  (10) 

'  fr~  •  (ID 


and 


By  means  of  equations  (6),  (7)  and  (8)  it  is  possible  to  calculate 
the  critical  constants  of  a  gas  when  the  constants  a  and  b  of  Van 
der  Waals'  equation  are  known.  If  we  take  carbon  dioxide  as  an 
example,  for  which  a  =  0.00874  and  b  =  0.0023  we  obtain 
vc  =  0.0069  (observed  value  =  0.0066),  pc  =  61  atmospheres, 
(observed  value  =  70  atmospheres),  Tc  =  305°.5  abs.,  (observed 
value  303°.  9  abs.).  Conversely  by  means  of  equations  (10)  and 
(11)  the  value  of  a  and  6  can  be  calculated  when  the  critical  data 
are  given. 

Corresponding  Conditions.  If  in  the  equation  of  Van  der 
Waals,  the  values  of  p}  v  and  T  be  expressed  as  fractions  of  the 
corresponding  critical  values,  we  may  write 

p  =  <xpc, 

v  =  (3vc 
and 

T  =  yTe. 


86  THEORETICAL  CHEMISTRY 

Substituting  these  values  in  the  equation 


we  have 

(<xpc  +  T^)(l3vc-b)=R>YTc, 


and  replacing  pc,  vc,  and  Tc  by  their  values  given  in  equations  (6), 
(7)  and  (8)  of  the  preceding  paragraph,  we  obtain 


which  is  Van  der  Waals'  reduced  equation  of  condition. 

In  this  equation  everything  connected  with  the  individual 
nature  of  the  substance  has  vanished,  thus  making  it  applicable 
to  all  substances  in  the  liquid  or  gaseous  state  in  the  same  way 
that  the  fundamental  gas  equation  is  applicable  to  all  gases  irre- 
spective of  their  specific  nature.  It  has  been  shown,  however, 
that  the  equation  is  not  entirely  trustworthy  and  at  best  can  be 
considered  as  little  more  than  a  rough  approximation.  The 
chief  point  to  be  observed  in  connection  with  this  equation  is 
that  whereas  for  gases,  the  corresponding  values  of  temperature, 
pressure  and  volume,  measured  in  the  ordinary  units,  may  be 
compared,  it  is  necessary  in  the  case  of  liquids  to  make  the  com- 
parison under  corresponding  conditions.  For  example,  the  molec- 
ular volumes  of  two  liquids  are  to  be  compared,  not  at  room 
temperature  but  at  temperatures  which  are  equal  fractions  of 
their  respective  critical  temperatures.  Such  temperatures  Van 
der  Waals  called  corresponding  temperatures. 

By  way  of  illustration,  suppose  we  wish  to  compare  alcohol 
and  ether  with  respect  to  some  particular  property,  such  as 
surface  tension.  If  the  surface  tension  of  alcohol  be  measured 
at  60°,  at  what  temperature  must  a  similar  measurement  be 
made  with  ether  in  order  that  the  results  may  be  comparable? 
The  critical  temperature  of  alcohol  is  243°  C.  or  516°  absolute; 
that  of  ether  is  194°  C.  or  467°  absolute.  Then  according  to  Van 
der  Waals'  definition  of  corresponding  conditions,  the  temperature, 


LIQUIDS 


87 


t,  at  which  measurements  should  be  made  with  ether  will  be  given 
by  the  proportion,  273  +  t  :  467  ::  273  +  60  :  516,  or  t  =  28°  C. 
By  making  comparisons  of  various  properties  at  corresponding 
temperatures  it  has  been  found  that  greater  regularities  are 
observed  than  when  comparisons  are  made  at  the  same  tempera- 
ture, thus  justifying  the  claim  of  Van  der  Waals. 

Liquefaction  of  Gases.  The  history  of  the  liquefaction  of 
gases  has  for  a  long  time  been  regarded  as  one  of  the  most 
interesting  chapters  of  physical  science.  Among  the  first  success- 


Fig.  24. 

ful  workers  in  this  field  was  Faraday.*  He  liquefied  practically 
all  of  the  gases  which  condense  under  moderate  pressures  and  at 
not  very  low  temperatures.  A  sketch  of  the  apparatus  used  by 
Faraday  is  shown  in  Fig.  24.  It  consisted  of  an  inverted 
V-shaped  tube,  in  one  end  of  which  was  placed  some  solid  which 
would  liberate  the  desired  gas  on  heating,  while  the  other  end 
was  sealed  and  immersed  in  a  freezing  mixture.  When  the  sub- 
stance at  A  had  been  heated  long  enough  to  liberate  considerable 
gas,  the  pressure  within  the  tube  became  sufficiently  high  to  cause 
the  gas  to  liquefy  at  the  temperature  of  the  end  B.  Thus  chlorine 
*  Phil.  Trans.  113,  189  (1823). 


88  THEORETICAL  CHEMISTRY 

hydrate  was  heated  in  the  tube  and  the  liberated  chlorine  was 
condensed  at  B  as  a  yellow  liquid.  In  1834,  Thilorier  *  succeeded 
in  liquefying  carbon  dioxide  in  quite  large  amounts  by  the  use  of 
a  new  form  of  apparatus.  In  connection  with  his  experiments 
on  liquid  carbon  dioxide,  he  observed  that  when  it  was  allowed  to 
vaporize,  enough  heat  was  absorbed  to  lower  the  temperature 
below  its  freezing  point,  solid  carbon  dioxide  being  obtained.  He 
discovered  that  a  mixture  of  solid  carbon  dioxide  and  ether  was  a 
powerful  refrigerant,  and  that  under  diminished  pressure  the 
mixture  gave  temperatures  ranging  from  —  100°  C.  to  —  110°  C. 
This  mixture  is  known  today  as  Thilorier' s  mixture.  Faraday  f 
undertook  the  liquefaction  of  the  so-called  permanent  gases  in  1845. 
In  this  second  series  of  experiments  by  Faraday  he  employed 
higher  pressures  than  in  his  earlier  experiments,  and  also  made 
use  of  the  newly  discovered  Thilorier  mixture  as  a  refrigerant. 
He  was  partially  successful  in  his  attempt  to  liquefy  the  hitherto 
noncondensible  gases.  He  liquefied  ethylene,  phosphine  and 
hydrobromic  acid  and  also  solidified  ammonia,  cyanogen,  and 
nitrous  oxide.  He  failed  to  liquefy  hydrogen,  oxygen,  nitrogen, 
nitric  oxide  and  carbon  monoxide.  No  further  advance  in  the 
liquefaction  of  gases  was  made  until  the  year  1869  when  Andrews 
pointed  out  the  importance  of  cooling  the  gas  below  its  critical 
temperature.  This  discovery  explained  why  so  many  of  the 
earlier  experiments  had  failed,  and  opened  the  way  to  the  brilliant 
successes  of  the  latter  part  of  the  nineteenth  century.  In  1887, 
Cailletet  J  and  Pictet,  §  working  independently,  succeeded  in 
liquefying  oxygen.  Cailletet  subjected  the  gas  to  a  pressure  of 
about  300  atmospheres  using  boiling  sulphur  dioxide  as  a  refriger- 
ant. The  gas  was  further  cooled  by  suddenly  releasing  the  pres- 
sure and  allowing  it  to  expand.  In  addition  to  oxygen  he  also 
liquefied  air,  nitrogen  and  possibly  hydrogen.  Shortly  afterward 
in  1883,  the  Polish  scientists,  Wroblewski  and  Olszewski,  1f  pub- 

*  Lieb.  Ann.,  30,  122  (1839). 
t  Phil.  Trans.,  135,  155  (1845). 
t  Compt.  rend.,  85,  1217  (1877). 
§  Ibid.,  85,  1214,  1220  (1877). 
H  Wied.  Ann.,  20,  243  (1883). 


LIQUIDS  89 

lished  an  account  of  their  interesting  and  highly  important  work. 
In  their  experiments  they  subjected  the  gas  to  be  liquefied  to  high 
pressure,  and  simultaneously  cooled  it  to  a  very  low  temperature. 
Among  the  refrigerants  used  by  them  was  liquid  ethylene,  which 
was  allowed  to  boil  off  under  diminished  pressure,  giving  a  temper- 
ature of  —  130°  C.  At  this  temperature,  a  pressure  of  only 
20  atmospheres  was  sufficient  to  condense  oxygen  to  the  liquid 
state.  Having  liquefied  oxygen,  nitrogen,  air  and  carbon  mon- 
oxide, and  having  determined  the  boiling-points  of  these  gases 
under  atmospheric  pressure,  they  proceeded  to  use  these  liquefied 
gases  as  refrigerants,  allowing  them  to  boil  off  under  diminished 
pressure,  thus  obtaining  temperatures  as  low  as  —  200°  C.  A 
very  small  amount  of  liquid  hydrogen  was  obtained  in  this  way. 
Subsequent  attempts  by  these  same  experimenters  to  liquefy 
hydrogen,  while  not  much  more  successful  than  their  former 
attempts,  enabled  them  to  determine  its  boiling-point.  Shortly 
after  the  publication  of  the  first  papers  of  Wroblewski  and 
Olszewski,  Dewar  *  devised  a  new  form  of  apparatus  for  lique- 
fying air,  oxygen  and  nitrogen  on  a  comparatively  large  scale. 
He  also  introduced  the  well-known  vacuum-jacketed  flasks  and 
tubes  which  greatly  facilitated  carrying  out  experiments  with 
liquefied  gases.  In  1895,  Linde  in  Germany  and  Hampson  in 
England  simultaneously  and  independently  constructed  machines 
for  the  liquefaction  of  air  in  large  quantities. 

In  the  method  devised  by  these  experimenters  the  air  is  not 
subjected  to  a  preliminary  cooling,  produced  by  the  rapid  evapora- 
tion of  a  liquefied  gas  under  diminished  pressure,  as  in  the  methods 
of  Wroblewski  and  Olszewski.  In  the  Linde  liquefier,  the  air  is 
compressed  to  about  200  atmospheres.  It  is  then  passed  through 
a  chamber  containing  anhydrous  calcium  chloride  to  remove  the 
greater  part  of  the  moisture,  after  which  it  is  cooled  by  allowing  it 
to  circulate  through  a  coiled  pipe  immersed  in  a  freezing  mixture. 
Nearly  all  of  the  moisture  remaining  in  the  air  is  deposited  on  the 
walls  of  the  pipe  in  the  form  of  frost.  The  air  then  enters  a  long 
spiral  tube  jacketed  with  a  non-conducting  material,  and  is  there 
allowed  to  expand  to  a  pressure  of  about  15  atmospheres. 

*  Prpc.  Roy.  Inst.,  1886,  550, 


90  THEORETICAL  CHEMISTRY 

During  this  expansion  the  temperature  of  the  air  is  appreciably 
lowered.  When  the  air  has  traversed  the  spiral  tube,  it  is  still 
further  cooled  by  allowing  it  to  expand  to  a  pressure  equal  to  that 
of  the  atmosphere.  The  air  which  has  been  thus  cooled  is  then 
passed  backward  through  the  annular  space  between  the  spiral  tube 
and  a  concentric  jacket,  thus  cooling  the  entering  portion  of  air. 
Consequently  this  next  portion  of  air  expands  from  a  lower  initial 
temperature,  and  the  cooling  effect  is  increased.  In  like  manner, 
when  this  cooler  air  passes  backward  it  cools  still  further  the  next 
succeeding  portion,  and  eventually  the  temperature  is  reduced 
sufficiently  to  cause  a  small  amount  of  the  air  to  liquefy  as  it 
issues  from  the  end  of  the  spiral  tube.  The  remaining  portion  of 
the  air  which  has  not  been  liquefied,  passes  backward  through  the 
annular  tube  and  cools  the  following  portion  to  a  still  greater 
extent,  causing  a  larger  proportion  to  liquefy  on  expansion.  With 
a  3-horse-power  machine,  a  continuous  supply  of  0.9  liter  per  hour 
can  be  obtained.  Further  improvements  in  this  apparatus  have 
been  made  by  Dewar  and  Hampson,  and  by  means  of  it  Dewar's 
brilliant  successes  in  the  liquefaction  of  gases  have  been  achieved. 
The  most  efficient  apparatus  for  the  liquefaction  of  air  and  other 
gases  is  that  developed  by  Claude.*  The  essential  features  of 
his  liquefier  are  shown  in  Fig.  25.  The  air  is  first  compressed  to 
40  atmospheres  pressure  by  means  of  an  ordinary  compression 
pump  not  shown  in  the  diagram,  the  moisture  and  carbon  dioxide 
being  removed  as  in  Linde's  method.  It  then  enters  the  tube  A, 
which  in  reality  is  of  a  spiral  form,  and  divides  at  B.  A  portion 
enters  the  cylinder  D  through  a  valve  chest  similar  to  that  in  a 
steam  engine  forces  out  the  piston  and  causes  the  wheel,  TF, 
to  revolve,  thereby  doing  work  and  cooling  the  air.  The  cooled 
air  escapes  from  the  valve  chest  and  circulates  through  the  lique- 
fying chamber  L,  where  it  causes  the  portion  of  compressed  air 
entering  at  B  to  liquefy.  It  then  issues  from  the  liquefier  and 
traverses  M,  cooling  the  entering  portion  of  air  in  A,  and  finally 
returns  to  the  compressor.  The  pressure  of  the  air  when  it  issues 
from  D  is  almost  atmospheric,  and  its  temperature  is  below 
—  140°  C.  About  twenty-five  per  cent  of  the  power  consumed 
*  Compt.  rend.,  II,  500  (1900);  I,  1568  (1902);  II,  762,  823,  (1905). 


LIQUIDS 


91 


in  compression  is  regained  by  the  motor.  The  apparatus  pro- 
duces about  1  liter  of  liquid  air  per  horse-power  hour.  By  means 
of  this  improved  apparatus,  based  upon  the  regenerative  principle, 
all  known  gases  have  now  been  liquefied,  the  last  to  succumb  being 


Liquid  Air       o 
,40  atmos..— 140 


Fig.  25. 


helium  which  was  liquefied  in  1908,  by  Kammerlingh  Onnes  in  the 
Leyden  cryogenic  laboratory.  The  subjoined  table  gives  the 
critical  data  together  with  the  boiling  and  freezing  temperatures 
of  some  of  the  more  common  gases. 

CRITICAL  DATA  FOR  GASES. 


Gas. 

Crit.  Temp. 

Crit.  Press. 
(Atmos.) 

Boiling  Point 
(at  760  mm.). 

Freezing 
Point. 

Helium  

-267° 

Hydrogen  
Air  
Nitrogen  

-242° 
-140° 
-146° 

20 
39 
35 

-252°.  5 
-191° 
-195°.  5 

-258°.  9 

-2i6°!5 

Oxygen  

-118° 

50.8 

-182°.  8 

-227° 

Carbon  monoxide  
Nitric  oxide  
Carbon  dioxide  
Hydrochloric  acid  
Ammonia     

-141° 
-  93°.  5 
+  31° 
+  51°.  3 
+130° 

36 
71.2 
75 
86 
115 

-190° 
-153°.  6 
-  78° 
-  35° 
-  33°.7 

-207° 
-167° 
-  65° 
-116° 

-  77° 

92  THEORETICAL  CHEMISTRY 

Vapor  Pressure  of  Liquids.  According  to  the  kinetic  theory 
there  is  a  continuous  flight  of  particles  of  vapor  from  the  surface 
of  a  liquid  into  the  free  space  above  it.  At  the  same  time  the 
reverse  process  of  condensation  of  vapor  particles  at  the  surface 
of  the  liquid  is  taking  place.  Eventually  a  condition  of  equilib- 
rium will  be  established  between  the  liquid  and  its  vapor,  when 
the  rate  of  escape  will  be  exactly  counterbalanced  by  the  rate  of 
condensation  of  vapor  particles.  The  pressure  exerted  by  the 
vapor  of  a  liquid  when  equilibrium  has  been  attained  is  known  as 
its  vapor  pressure.  The  equilibrium  between  a  liquid  and  its  vapor 
is  dependent  upon  the  temperature.  For  every  temperature 
below  the  critical  temperature,  there  is  a  certain  pressure  at  which 
vapor  and  liquid  may  exist  in  equilibrium  in  all  proportions;  and 
conversely  for  every  pressure  below  the  critical  pressure,  there  is 
a  certain  temperature  at  which  vapor  and  liquid  may  exist  in 
equilibrium  in  all  proportions.  This  latter  temperature  is  termed 
the  boiling-point  of  the  liquid.  The  vapor  pressure  of  a  liquid 
may  be  measured  directly  by  placing  a  portion  of  it  above  the 
mercury  in  the  vacuum  of  a  barometer  tube,  heating  to  the  desired 
temperature,  and  observing  the  depression  of  the  mercury  column. 
This  is  known  as  the  static  method.  It  is  open  to  the  objection 
that  the  presence  of  volatile  impurities  in  the  liquid  causes  too 
great  depression  of  the  mercury  column,  the  vapor  pressure  of 
the  impurity  adding  itself  to  that  of  the  liquid  whose  vapor 
pressure  is  sought.  A  better  method  for  the  measurement  of 
vapor  pressure  is  that  known  as  the  dynamic  method.  In  this 
method  the  pressure  is  maintained  constant  and  the  boiling 
temperature  is  determined  with  an  accurate  thermometer.  The 
boiling  temperatures  corresponding  to  various  pressures  may  be 
measured,  provided  we  have  a  suitable  device  for  changing  and 
measuring  the  pressure.  The  results  obtained  by  the  static  and 
dynamic  methods  agree  closely  if  the  liquid  is  pure,  but  if  volatile 
impurities  are  present  the  results  obtained  by  the  dynamic  method 
are  more  trustworthy.  A  method  for  the  measurement  of  vapor 
pressure  due  to  James  Walker  *  is  of  considerable  interest.  In 
this  method,  a  current  of  pure  dry  air  is  bubbled  through  a  weighed 
*  Zeit.  phys.  Chem.,  2,  602  (1888). 


LIQUIDS  93 

amount  of  the  liquid  whose  vapor  pressure  is  to  be  determined. 
The  liquid  is  maintained  at  constant  temperature  and  its  loss  in 
weight  is  observed.  In  passing  through  the  liquid  the  air  will 
absorb  an  amount  of  vapor  directly  proportional  to  the  vapor 
pressure  of  the  liquid.  If  1  mol  of  liquid  is  absorbed  by  v  liters 
of  air,  then  we  have 

pv  =  RT, 

where  p  is  the  vapor  pressure  of  the  liquid,  and  T  its  temperature. 
If  Vi  is  the  volume  of  air  which  absorbs  g  grams  of  the  vapor  of 
the  liquid  whose  molecular  weight  is  M,  then 


or 


In  this  equation,  Vi  denotes  the  total  volume  containing  g  grams 
of  the  liquid  in  the  form  of  vapor,  or  in  other  words  it  represents 
the  air  and  vapor  together.  Since  the  volume  of  the  air  is  in 
general  so  much  greater  than  that  of  the  vapor,  Vi  may  be  taken 
as  that  of  the  air  alone. 

Heat  of  Vaporization.  In  order  to  transform  a  liquid  into  a 
vapor  a  large  amount  of  heat  is  required.  Thus,  when  a  liquid 
is  heated  to  the  boiling-point,  the  volume  must  be  increased  against 
the  pressure  of  the  atmosphere,  external  work  being  done,  and 
when  the  boiling  temperature  is  reached  the  liquid  must  be  vapor- 
ized; the  heat  expended  in  causing  the  change  of  physical  state 
being  much  greater  than  that  required  to  expand  the  liquid.  An 
interesting  relation  between  the  heat  of  vaporization  and  the 
absolute  boiling-point  of  a  liquid  was  discovered  by  Trouton.* 
If  T  denotes  the  absolute  boiling-point  and  w  the  heat  of  vapor- 
ization of  1  gram  of  liquid  whose  molecular  weight  is  M,  then 
according  to  Trouton 

Mw  _ 
~Y  -21, 

or  in  words,  the  ratio  of  the  molecular  heat  of  vaporization  to  the 
absolute  boiling  temperature  of  a  liquid  is  constant,  the  numerical 

*  Phil.  Mag.  (5),  18,  54  (1884). 


94 


THEORETICAL  CHEMISTRY 


value  of  the  ratio  being  approximately  21.  This  is  known  as 
Trouton's  law.  While  this  relation  holds  quite  well  for  many 
liquids,  Nernst  has  pointed  out  that  the  constant  varies  with  the 
temperature,  and  has  proposed  two  other  forms  of  the  equation. 
Bingham  has  simplified  the  equations  of  Nernst  to  the  following 
form  :  — 


=  17+0.011  T. 

While  this  modification  of  the  Trouton  equation  has  been  found 
to  hold  for  a  large  number  of  substances,  there  are  other  substances 
for  which  the  left  side  of  the  equation  has  a  value  greater  than 
that  of  the  right  side.  Bingham  infers  that  where  this  occurs,  the 
substance  in  the  liquid  state  has  a  greater  molecular  weight 
than  it  has  in  the  gaseous  state,  or  in  other  words,  the  liquid  is 
associated.  It  is  evident  that  an  associated  liquid  will  require 
an  expenditure  of  energy  over  and  above  that  required  for  vapori- 
zation, to  break  down  the  molecular  complex.  The  difference 
between  the  values  of  the  two  sides  of  the  equation  may  be 
taken  as  a  rough  measure  of  the  degree  of  association. 

Boiling-Point  and  Critical  Temperature.  An  interesting  rela- 
tion has  been  pointed  out  by  Guldberg  *  and  Guye.f  These 
two  investigators  have  shown  that  the  absolute  boiling  temper- 
ature of  a  liquid  is  about  two-thirds  of  its  critical  temperature. 
That  this  empirical  relation  holds  for  a  variety  of  different  sub- 
stances is  shown  in  the  accompanying  table. 

RELATION   OF  BOILING-POINT  TO  CRITICAL  TEMPERATURE. 


Substance. 

Tb 

Te 

Tb/Tc 

Oxygen 

90° 

155° 

0.58 

Chlorine 

240° 

414° 

0.58 

Sulphur  dioxide  . 

263° 

429° 

0.61 

Ethyl  ether        .  .           

308° 

467° 

0.66 

Ethyl  alcohol  

351° 

516° 

0.68 

Benzene  

353° 

562° 

0.63 

Water  

373° 

637° 

0.59 

Phenol 

454° 

691° 

0.66 

*  Zeit.  phys.  chem.,  5,  376  (1890). 
t  Bull.  Soc.  Chim.,  (3),  4,  262  (1890). 


LIQUIDS  95 

Molecular  Volume.  In  dealing  with  the  volume  relations  of 
liquids  it  is  customary  to  employ  the  molecular  volume,  i.e.,  the 
volume  occupied  by  the  molecular  weight  of  the  liquid  in  grams. 
The  justification  of  this  procedure  is  that  when  we  compare  the 
gram-molecular  weights  of  liquids,  the  comparison  involves  equal 
numbers  of  molecules  of  the  different  substances.  Since 


we  may  write 
and  similarly, 


,  mass 

volume  =  -5 r— ) 

density 


molecular  weight 

molecular  volume  = 3 T— -    - — , 

density 


.  atomic  weight 

atomic  volume  = ^ r—  ^— 

density 


Relations  between  the  molecular  volumes  of  liquids  were  first 
pointed  out  by  Kopp.*  On  comparing  the  molecular  volumes  of 
different  liquids  at  their  boiling-points,  he  found  that  constant 
differences  in  composition  correspond  to  constant  differences  in 
the  molecular  volumes.  Thus  the  molecular  volumes  of  the 
successive  members  of  an  homologous  series  differ  by  the  same 
number  of  units,  this  difference  corresponding,  for  example,  to  a 
CH2  group.  In  like  manner  the  molecular  volumes  of  various 
groups  have  been  determined,  and  from  these  in  turn  the  atomic 
volumes  of  the  constituent  elements  have  been  worked  out.  The 
atomic  volumes  assigned  by  Kopp  to  some  of  the  elements  com- 
monly entering  into  organic  compounds  are  as  follows :  — 

C  =  11         Cl=22.8  1=37.5       Hydroxyl  0  =  7 . 8 

#  =  5.5       Br  =  27.8  S  =  22.6      Carbonyl  0  =  12.2 

The  value  of  the  atomic  volume  is  found  to  be  dependent  upon 
the  manner  of  linkage;  thus  oxygen  in  the  hydroxyl  group  has  the 
atomic  volume,  7.8,  while  oxygen  in  the  doubly  linked  condi- 
tion, as  in  the  carbonyl  group,  has  the  atomic  volume,  12.2.  By 
means  of  such  a  table  of  experimentally  determined  atomic 

*  Lieb.  Ann.,  41,  79  (1842);  96,  153,  303  (1855);  96,  171  (1855). 


96  THEORETICAL  CHEMISTRY 

volumes,  Kopp  showed  that  it  is  possible  to  calculate  the  molec- 
ular volume  of  a  liquid  with  a  fair  degree  of  accuracy.  For 
example,  the  molecular  volume  of  acetic  acid  C2H4O2  may  be 
calculated  from  the  atomic  volumes  of  its  constituent  atoms  as 
follows:  — 

20  =  2X11=      22 

4H  =  4  X  5.5  =    22 
IHydroxylO  =  1  X  7.8  =      7.8 
1  Carbonyl  0  =  1  X  12.2  =  12.2 
Molecular  volume  =  64 . 0 

The  density  of  acetic  acid  at  its  boiling-point  is  0.942,  and  its 
molecular  weight  is  60,  therefore  the  observed  value  of  the  molec- 
ular volume  is  60  -5-  0.942  =  63.7,  a  result  which  is  in  excellent 
agreement  with  that  calculated  from  the  atomic  volumes  of  the 
constituents.  The  more  recent  investigations  of  Thorpe,  Lossen, 
Schiff  and  Buff  afford  a  confirmation  of  the  conclusion  reached  by 
Kopp,  that  the  molecular  volumes  of  liquids  are  in  general  additive. 
While  Kopp  found  that  his  results  were  most  regular  when  the 
molecular  volumes  were  determined  at  the  boiling  temperatures 
of  the  respective  liquids,  the  reason  for  this  did  not  appear  until 
after  Van  der  Waals  had  developed  his  theory  of  corresponding 
states.  As  has  been  pointed  out  in  the  preceding  paragraph  the 
boiling-points  of  most  liquids  are  approximately  two-thirds  of 
their  respective  critical  temperatures,  and  therefore  the  boiling- 
points  are  corresponding  temperatures. 

Co-volume.  By  studying  various  series  of  hydrocarbons, 
alcohols  and  ethers,  Traube  *  has  been  led  to  suggest  that  the 
molecular  volume  of  a  liquid  be  looked  upon  as  made  up  of  the 
atomic  volumes  of  its  constituent  elements  and  a  magnitude 
which  he  terms  the  co-volume.  This  latter  he  defines  as  the  space 
surrounding  a  molecule  within  which  it  is  free  to  vibrate  and  from 
which  other  molecules  are  excluded.  The  co-volume  appears  to 
be  nearly  constant  for  a  large  number  of  substances,  its  mean 
value  at  a  temperature  of  15°  C.  being  25.9  cc.  The  values 

*  Uber  den  Raum  der  Atome.  J.  Traube.  Ahrens'  Sammlung  Chemischer 
und  chemisch-technischer  Vortraege,  4,  255  (1899). 


LIQUIDS  97 

assigned  by  Traube  to  the  atomic  volumes  of  some  of  the  elements 
are  as  follows :  — 

C  =  9.9     0  =    5.5       Br  =  17.7     N  (trivalent)       =    1.5 
H  =  3.1     Cl  =  13.2  1=  21.4     N  (pentavalent)  =  10.7 

Traube  has  worked  out  a  series  of  constants  which  must  be  de- 
ducted to  allow  for  ring  formation  and  for  double  and  treble  link- 
ing. By  means  of  these  values,  it  is  possible  to  calculate  the 
molecular  volume  of  a  substance  by  adding  together  the  respective 
atomic  volumes  of  the  constituents  of  the  liquid  and  the  co- 
volume,  25.9.  It  is  of  course  necessary  to  know  the  molecular 
weight  of  the  substance  together  with  its  constitution,  so  that 
due  allowance  may  be  made  for  unsaturation.  For  example, 
the  molecular  volume  of  ethyl  ether,  C^ioO,  may  be  calculated 
by  Traube's  method  as  follows :  — 

4C=    4X9.9  =  39.6 
10H  =  10  X  3.1  =  31 
10=     1  X5.5  =    5.5 
76.1 

Co-volume    25 . 9 
Molecular  volume  102.0 

The  molecular  volume,  as  determined  from  the  molecular  weight 
and  density  at  15°  C.,  is  74  -f-  0.7201  =  102.7. 

The  method  of  Traube  may  be  employed  in  roughly  checking 
the  accepted  value  of  the  molecular  weight  of  a  liquid  provided 
its  density  at  15°  C.  is  known,  since  in  the  equation 

M/d  =  S  atomic  volumes  +  25.9,  expressing  Traube's  relation, 

M  is  the  only  unknown  quantity.  It  is  apparent  that  the  liquid 
must  be  non-associated,  since  for  an  associated  substance  the 
normal  co-volume  must  necessarily  accompany  the  polymerized 
molecule.  In  this  case  the  formula  becomes 

M/d  =  2  atomic  volumes  +  25.9/n, 

where  n  denotes  the  number  of  simple  molecules  in  the  polymer. 
Obviously  when  the  molecular  weight  of  a  liquid  is  known,  the 
experimental  determination  of  the  co-volume,  (M/d  —  S  atomic 
vols.)  may  be  used  to  estimate  the  degree  of  association.  The 


98 


THEORETICAL  CHEMISTRY 


values  thus  obtained  are  not  in  satisfactory  agreement  with  the 
factors  of  association  derived  by  means  of  other  methods. 

Refractive  Power  of  Liquids.     The  velocity  of  transmission 
of  light  through  any  medium  depends  upon  its  nature,  especially 


Fig.  26. 

upon  its  density.  When  a  ray  of  light  passes  from  one  medium 
into  another  it  is  refracted,  the  degree  of  refraction  being  such 
that  the  ratio  of  the  sines  of  the  angles  of  incidence  and  refrac- 
tion is  constant  and  characteristic  for  the  two  media.  This 


LIQUIDS 


99 


fundamental  law  of  refraction  was  discovered  by  Snell  about 
1621.  According  to  the  wave  theory  of  light,  the  ratio  of  the 
sines  of  the  angles  of  incidence  and  refraction  is  identical  with 
the  ratio  of  the  velocities  of  light  in  the  two  media.  The  ratio  is 
termed  the  index  of  refraction  and  is  usually  denoted  by  the  letter 
7i.  Representing  by  i  and  r,  the  angles  of  incidence  and  refraction, 
and  by  v\  and  v2,  the  respective  velocities  of  light  in  the  two  media, 
we  have 

sin  i      Vi 


n 


sm  r 


Various  forms  of  apparatus  have  been  devised  for  the  determina- 
tion of  the  refractive  index  of  liquids.  Of  these  the  best  known 
and  most  satisfactory  is  the  refractometer  of  Pulfrich,  an  unproved 
form  of  which  is  shown  in  Fig.  26.  While  the  limits  of  this  book 
prohibit  a  detailed  description  of  the  apparatus,  the  fundamental 
principles  involved  in  its  construction  will  be  readily  understood 
from  the  accompanying  diagram,  Fig.  27.  The  liquid  or  fused 
solid  is  placed  in  a  small  glass  cell,  C,  which  is  cemented  to  a  rec- 
tangular prism  of  dense  optical 
glass,  P,  the  refractive  index  of 
which  is  generally  1.61.  A  beam 
of  monochromatic  light,  from  a 
sodium  flame  or  a  spectrum-tube 
containing  hydrogen,  is  allowed  to 
enter  the  prism  in  a  direction  par- 
allel to  the  horizontal  surface  of 
separation  between  the  glass  and 
the  liquid.  After  passing  through 


Fig.  27. 


the  liquid  and  the  prism,  the  beam  emerges  making  an  angle  i  with 
its  original  direction.  By  means  of  a  telescope,  the  emergent  beam 
can  be  observed  and  its  position  noted,  the  angle  of  emergence 
being  read  on  a  divided  circle  attached  to  the  telescope.  From  the 
angle  of  emergence  thus  determined,  the  index  of  refraction  of  the 
liquid  can  be  calculated  in  the  following  manner.  The  value  of 
the  index  of  refraction,  M ,  for  air/glass  being  known,  we  have 


smi 
sinr 


(1) 


100  THEORETICAL  CHEMISTRY 

The  angle  of  incidence  of  the  light  entering  the  prism  from  the 
liquid  is  90°,  or  sin  i\  =  1.  The  index  of  refraction,  HI,  for  liquid/ 
glass  may  be  calculated  thus, 


sm  TI      sin  7*1 
But 


_ 

sin  ri  =  cosr  =  V  1  —  sin2r.  (3) 

Transposing  equation  (1)  and  substituting  in  equation  (3),  we 
have 


I  —  sin2  i 


or 

sin  ri  =  rr=  VN2  —  sin2  i.  (4) 

Therefore,  substituting  equation  (4)  in  equation  (2),  we  have 

N 
^'  (5) 

Remembering  that  n  =  N/rii,  we  have  for  the  index  of  refraction, 
n,  for  air /liquid,  by  substitution  in  equation  (5), 


-  sirfi, 


or  if  N  =  1.61, 


n  =  V2.5921  -  sin2^. 


The  values  of  VjV2  —  sin2  i  are  generally  given,  for  different  values 
of  i,  in  tables  supplied  with  the  refractometer,  thus  saving  the 
experimenter  a  somewhat  laborious  calculation.  The  value  of  n 
thus  obtained  is  the  index  of  refraction  from  air  into  the  liquid;  if 
the  index  from  vacuum  into  the  liquid,  the  so-called  absolute  index, 
is  required,  the  value  of  n  must  be  multiplied  by  1.00029.  The 
index  of  refraction  is  dependent  upon  temperature,  pressure  and 
in  general  upon  all  conditions  which  affect  the  density  of  the 
medium.  Furthermore,  it  is  dependent  upon  the  wave-length  of 
the  light  employed,  the  index  for  the  red  rays  being  greater  than 
that  for  the  violet  rays.  It  is  therefore  necessary  in  making 


LIQUIDS  101 

measurements  of  refractive  indices  to  use  light  of  a  definite  wave- 
length, or  what  is  termed  monochromatic  light.  The  sodium 
flame  is  most  frequently  used  for  this  purpose,  the  wave-length 
being  represented  by  the  letter  D.  Measurements  of  the  refrac- 
tive index  referred  to  the  D-line  of  sodium  are  commonly  desig- 
nated by  the  symbol  UD>  When  incandescent  hydrogen  is  employed 
as  a  source  of  light,  the  refractive  index  may  be  determined 
for  the  C-,  F-  and  (r-lines,  the  respective  values  being  represented 
by  nc,  np,  and  no- 

Specific  and  Molecular  Refraction.  Various  attempts  have 
been  made  to  express  the  refractive  power  of  a  liquid  by  a  formula 
which  is  independent  of  variations  of  temperature  and  pressure. 
Of  the  different  formulas  proposed  but  two  need  be  mentioned. 
The  first,  due  to  Gladstone  and  Dale,*  is  as  follows:  — 


in  which  d  denotes  the  density  of  the  liquid  and  TI  is  the  so-called 
specific  refraction.  The  other  formula,  proposed  by  Lorenz  |  and 
Lorentz,J  has  the  following  form:  — 

1   n2-  1 
Tz      d'n*  +  2' 

This  formula  is  superior  to  that  of  Gladstone  and  Dale  which  is 
purely  empirical.  It  is  based  upon  the  electromagnetic  theory 
of  light  and  gives  values  of  r2  which  are  quite  independent  of  the 
temperature.  In  order  that  we  may  compare  the  refractive 
powers  of  different  liquids,  the  specific  refractions  are  multiplied 
by  their  respective  molecular  weights,  the  resulting  products 
being  termed  their  molecular  refractions.  As  the  result  of  a  large 
number  of  experiments,  it  has  been  shown  that  the  molecular 
refraction  of  a  compound  is  made  up  of  the  sum  of  the  refractive 
constants  of  the  constituent  atoms,  or  in  other  words  refractive 
power  is  an  additive  property.  The  values  of  the  refractive  con- 
stants of  the  elements  and  commonly-occurring  groups  have  been 

*  Phil.  Trans.  (1858). 

t  Wied.  Ann.,  n,  70  (1880). 

t  Ibid.,  9,  641  (1880). 


102  THEORETICAL  CHEMISTRY 

determined  with  great  care  by  Briihl  and  others,  the  method 
employed  being  similar  to  that  used  by  Kopp  in  connection  with 
his  investigations  on  molecular  volumes.  Thus,  Briihl  found  in 
the  homologous  series  of  aliphatic  compounds  that  a  difference 
of  CH2  in  composition  corresponds  to  a  constant  difference  of 
4.57  hi  molecular  refraction.  Then,  having  determined  the 
molecular  refraction  of  a  ketone  or  an  aldehyde  of  the  composi- 
tion, CnH2nO,  he  subtracted  n  times  the  value  of  CH2  and  the 
atomic  refraction  of  carbonyl  oxygen.  By  deducting  the  molecular 
refraction  of  the  hydrocarbon,  CnH2n+2,  from  that  of  the  corre- 
sponding alcohol,  CnHzn+zO,  he  obtained  the  atomic  refraction  of 
hydroxyl  oxygen.  By  subtracting  six  times  the  value  of  CH2 
from  the  molecular  refraction  of  hexane,  C6Hi4,  he  obtained  the 
refractive  constant  for  hydrogen  or  2  H  =  2.08.  In  like  manner 
the  refractions  of  other  elements  and  groups  of  elements  were 
determined. 

Just  as  in  the  case  of  molecular  volumes  so  with  molecular 
refractions,  the  arrangement  of  the  atoms  in  the  molecule  must 
be  taken  into  consideration.  Briihl,*  who  has  devoted  much  time 
to  the  investigation  of  the  effect  of  constitution  upon  refraction, 
has  pointed  out  that  the  molecular  refraction  of  compounds  con- 
taining double  and  triple  bonds  is  greater  than  the  calculated  value, 
and  he  has  assigned  to  these  bonds  definite  constants  of  refrac- 
tion. The  values  of  the  atomic  refractions  for  a  few  of  the  elements 
as  given  by  Briihl  are  as  follows :  — 

C=    2.48  Hydroxyl  0    =1.58 

H  =    1.04  Carbonyl  O    =  2.34 

Cl  =    6 . 02  Double  bond  =  1 . 78 

I  =  13 . 99  Triple  bond    =2.18 

More  recent  investigations  bring  out  the  fact  that,  when  double 
or  triple  bonds  occupy  adjacent  positions  in  the  molecule,  the 
simple  additive  relations  no  longer  obtain.  The  determination 
of  the  molecular  refraction  of  a  liquid  affords  a  means  of  ascertain- 
ing or  confirming  its  chemical  constitution.  For  example,  geraniol 
has  the  formula  Ci0Hi80,  and  its  chemical  behavior  is  such  as  to 

*  Proc.  Roy.  Inst.,  18,  122  (1906). 


LIQUIDS  103 

warrant  the  conclusion  that  it  is  a  primary  alcohol.  The  value 
of  UD  is  1.4745,  from  which  the  molecular  refraction  is  calcu- 
lated to  be  48.71.  The  molecular  refraction  calculated  from  the 
atomic  refractions  given  in  the  preceding  table  is  :  — 

IOC  =  10X2.48  =  24.80 

18H  =  18  X  1.04  =  18.72 

1  Hydroxyl  0  =    1  X  1.58  =    1.58 

Molecular  refraction  45  .  10 

The  difference  between  the  theoretical  and  experimental  values 
of  the  molecular  refraction  is  48.71  —  45.10  =  3.61,  which  is 
approximately  twice  the  value  of  a  double  bond,  1.78  X  2  =  3.56. 
From  this  we  conclude  that  the  molecule  of  geraniol  contains  two 
double  bonds.  Furthermore  an  alcohol  of  the  formula,  doHi80, 
containing  two  double  bonds  cannot  possess  a  ring  structure  and 
therefore  must  be  a  member  of  the  aliphatic  group  of  compounds. 
This  conclusion  is  supported  by  the  chemical  properties  of  the 
substance.*  In  a  similar  manner  the  Kekule"  formula  for  benzene 
has  been  confirmed,  the  difference  between  the  theoretical  and 
experimental  values  of  the  molecular  refraction  indicating  the 
presence  of  three  double  bonds  in  the  molecule. 

Specific  Refraction  of  Mixtures.  The  specific  refraction  of  an 
homogeneous  mixture  or  solution  is  the  mean  of  the  specific 
refractions  of  its  constituents.  Thus,  if  the  specific  refractions 
of  the  mixture  and  its  two  components  are  represented  by  rj,  r2, 
and  r3,  then 

p  (100  -p) 


where  p  denotes  the  percentage  of  the  constituent  whose  specific 
refraction  is  r2.  Hence  it  is  possible  to  determine  the  specific 
refraction  of  a  substance  in  solution  by  measuring  the  refractive 
indices  and  densities  of  the  solution  and  solvent.  If  the  refrac- 
tive indices  of  the  solvent,  solution  and  dissolved  substance  are 

*  The  accepted  structural  formula  of  geraniol  is 

H  -  C  -  CH2OH 
(CH8)2C  =  CH.CH2-CH2  -  C  -  CH, 


104  THEORETICAL  CHEMISTRY 

represented  by  n\,  n2,  and  n$  respectively,  and  if  d\,  dz,  and  d3 
denote  the  corresponding  densities,  then  we  have 

1  [  n32  -  1  =  100  nz2  -  1       100  -  p  n?-l 
d3"rc32  +  2  ~  d2p  *n22  +  2          dip      *n22  +  2> 

where  p  is  the  percentage  of  the  dissolved  substance.  As  has 
already  been  mentioned,  the  formula  of  Lorenz-Lorentz  is  based 
upon  the  electromagnetic  theory  of  light.  According  to  this 
theory  n2  —  l/n2  +  2  expresses  the  fraction  of  the  unit  of  volume 
of  the  substance  which  is  actually  occupied  by  it.  From  this  it 

M  n2  —  I 
follows  that  the  molecular  refraction,  -7-  •   2         ,  is  an  expression 

a     Tl    ~\~  £ 

of  the  volume  actually  occupied  by  the  atomic  nuclei  of  the 
molecule.  It  is  interesting  to  note  that  the  ratio  of  the  sum  of 
the  atomic  volumes,  calculated  by  the  method  of  Traube,  to  the 
corrected  molecular  volume,  as  determined  by  the  Lorenz-Lorentz 
formula,  is  approximately  constant,  or 

£  atomic  volumes     0  .  _ 

-  77  —  2  _  .  -  =3.45  approximately. 


This  may  be  considered  as  the  ratio  of  the  volume  within  which 
the  atoms  execute  their  vibrations  to  their  actual  material  volume. 
Rotation  of  the  Plane  of  Polarized  Light.  Some  liquids  when 
placed  in  the  path  of  a  beam  of  polarized  light  possess  the  prop- 
erty of  rotating  the  plane  of  polarization  to  the  right  or  to  the 
left.  Such  liquids  are  said  to  be  optically  active.  Those  substances 
which  rotate  the  plane  of  polarization  to  the  right  are  termed 
dextro-rotary,  while  those  which  cause  an  opposite  rotation  are 
called  levo-rotatory.  The  determination  of  the  rotatory  power  of 
a  liquid  is  made  by  means  of  an  instrument  known  as  a  polarimeter, 
a  convenient  form  of  which  is  shown  in  Fig.  28.  The  essential 
parts  of  this  instrument  are  two  similar  Nicol  prisms  placed  one 
behind  the  other  with  their  axes  in  the  same  straight  line.  The 
light  after  passing  through  the  forward  prism,  P,  known  as  the 
polarizer,  has  its  vibrations  reduced  to  a  single  plane;  it  is  said 
to  be  plane  polarized.  On  entering  the  rear  Nicol  prism,  A, 


LIQUIDS 


105 


known  as  the  analyzer,  the  light  will  either  pass  through  or  be 
completely  stopped,  depending  upon  the  position  of  the  prism. 
If  the  analyzer  be  slowly  rotated,  it  will  be  observed  that 
the  positions  of  maximum  transmission  and  extinction  occur  at 
points  90°  apart.  If  the  analyzer  be  rotated,  so  that  its  axis  is 
at  right  angles  to  the  axis  of  the  polarizer,  the  field  observed  will 
be  dark,  no  light  being  transmitted.  If  now  a  tube  similar  to 
that  shown  in  Fig.  29  be  filled  with  an  optically  active  liquid 
and  placed  between  the  polarizer  and  analyzer,  the  field  will 


Fig.  28. 

become  light  again,  due  to  the  rotation  of  the  plane  of  polariza- 
tion by  the  optically-active  substance.  The  extent  to  which  the 
plane  of  polarization  has  been  rotated  can  be  determined  by 
turning  the  analyzer  until  the  field  becomes  dark  again,  and  read- 
ing on  the  divided  circle,  K,  the  number  of  degrees  through  which 
it  has  been  moved.  When  it  is  necessary  to  turn  the  analyzer 
to  the  right,  the  substance  is  dextro-rotatory,  and  when  it  is  neces- 
sary to  turn  it  to  the  left,  the  substance  is  levo-rotatory.  Various 


106  THEORETICAL  CHEMISTRY 

optical  accessories  have  been  added  to  the  simple  polarimeter 
described  above  to  render  the  instrument  more  sensitive,  but  for 
these  details  the  student  must  consult  some  special  treatise.* 
The  angle  of  rotation  is  dependent  upon  the  nature  of  the  liquid, 
the  length  of  the  column  of  substance  through  which  the  light 
passes,  the  wave-length  of  the  light  used,  and  the  temperature  at 
which  the  measurement  is  made.  It  is  customary  in  polarimetric 


Fig.  29. 

work  to  employ  sodium  light  and,  unless  otherwise  specified,  it 
may  be  assumed  that  a  given  rotation  corresponds  to  the  D-line. 

Specific  and  Molecular  Rotation.  The  results  of  polarimetric 
measurements  are  expressed  either  as  specific  rotations  or  as 
molecular  rotations,  the  latter  being  preferable  since  the  optical 
activities  of  different  substances  may  then  be  compared. 

The  specific  rotation  is  obtained  by  dividing  the  observed  rota- 
tion by  the  product  of  the  length  of  the  column  of  liquid  and  its 
density,  or 

[a]t  =  w 

where  [a]t  is  the  specific  rotation  at  the  temperature,  t,  a  the 
observed  angle,  I,  the  length  of  the  column  of  liquid  in  decimeters 
and  d,  its  density.  If  the  specific  rotation  is  multiplied  by  the 
molecular  weight  of  the  substance  the  molecular  rotation  is  ob- 
tained, but  owing  to  the  fact  that  the  resulting  numbers  are  too 
large,  it  is  customary  to  express  the  molecular  rotation  as  one 
one-hundredth  of  this  value,  thus 

Ma 


The  specific  and  molecular  rotations  of  solutions  of  optically 
active  substances  may  also  be  determined,  if  we  assume  that  the 

*  See  for  example,  "The  Optical  Rotatory  Power  of  Organic  Substances 
and  its  Practical  Applications."    H.  Landolt,  trans,  by  J.  H.  Long. 


LIQUIDS 


107 


solvent  is  without  effect.  While  this  assumption  is  justifiable 
with  aqueous  solutions,  it  is  not  so  when  non-aqueous  solvents  are 
used.  If  g  grams  of  an  optically  active  substance  be  dissolved  in 
v  cc.  of  solvent,  then 

M 


\  i        av 
Mi  =  T: 


and 


-av 


or  if  the  composition  of  the  solution  is  expressed  in  terms  of 
weight  instead  of  volume,  g  grams  of  substance  being  dissolved 
in  100  grams  of  solution  of  density  d,  then 


100  a. 

gdl 


and     [aM\t  — 


Ma 

gdl 


Optical  Activity  and  Chemical  Constitution.  The  fact  that 
some  substances  have  the  power  of  rotating  the  plane  of  polarized 
light  was  first  discovered  by  Biot,  but  the  credit  for  recognizing 
the  chemical  significance  of  this  fact  belongs  to  Pasteur.*  He 
discovered  that  ordinary  recemic  acid  can  be  separated  into  two 
optically  active  modifications,  one  of  which  is  dextro-  and  the 
other  levo-rotatory,  the  numerical  values  of  the  two  rotations 


Fig.  30. 

being  identical.  If  a  solution  of  sodium  ammonium  racemate 
be  allowed  to  evaporate  at  a  low  temperature,  crystals  of  the 
composition  NaNH^H^Oe  •  4  H20  will  separate.  On  close  in- 
spection it  will  be  found  that  the  crystals  are  not  all  alike,  but 
that  they  may  be  divided  into  two  classes,  one  class  showing 
some  unsymmetrical  crystal  surfaces  which  are  oppositely  placed 
in  the  crystals  of  the  other  class.  The  crystals  of  one  class  may 


Ann.  Chim.  Phys.  (3),  24,  442  (1848);  28,  56  (1850);  31,  67  (1851). 


108  THEORETICAL  CHEMISTRY 

be  regarded  as  the  mirror  images  of  those  of  the  other  class: 
such  crystals  are  said  to  be  enantiomorphous.  The  forms  usually 
assumed  by  the  two  enantiomorphous  modifications  of  sodium 
ammonium  reacemate  are  shown  in  Fig.  30.  After  separating 
the  two  forms  Pasteur  dissolved  each  in  water,  making  the  solu- 
tions of  the  same  strength.  The  solution  of  the  crystals  with  the 
"right-handed  faces"  was  found  to  be  dextro-rotatory,  while  that 
of  the  crystals  with  the  "left-handed  faces"  was  found  to  be  levo- 
rotatory.  Pasteur  then  decomposed  the  two  salts  obtained  from 
sodium  ammonium  racemate  and  obtained  the  corresponding 
acids,  which  he  called  dextro-  and  levo-racemic  acids.  It  was 
subsequently  shown  that  the  two  acids  were  identical  with  dextro- 
and  levo-tartaric  acids.  Finally,  when  Pasteur  mixed  equiv- 
alent amounts  of  concentrated  solutions  of  dextro-and  levo- 
tartaric  acids,  an  appreciable  evolution  of  heat  was  observed, 
indicating  that  a  chemical  reaction  had  taken  place.  After 
allowing  the  solution  to  stand  for  some  time,  crystals  of  ordinary 
racemic  acid  were  obtained.  Thus  it  was  clearly  proven  that  an 
optically  inactive  substance  may  be  separated  into  two  opti- 
cally active  modifications,  possessing  equal  and  opposite  rotatory 
powers,  and  that  by  mixing  equivalent  quantities  of  the  two 
optically  active  forms,  the  optically  inactive  substance  may  be 
recovered. 

Pasteur  discovered  and  applied  three  other  methods  in  addi- 
tion to  the  mechanical  method  already  described,  for  the  separation 
of  a  substance  into  its  optically  active  modifications.  These  are 
as  follows:  —  (a)  Method  of  Crystallization',  (b)  Method  of  Forma- 
tion of  Derivatives;  and  (c)  Method  of  Ferments. 

Method  of  Crystallization.  To  a  supersaturated  solution  of  the 
racemic  modification  a  very  small  crystal  of  one  of  the  active 
forms  is  added.  This  will  induce  the  separation  of  crystals  of 
the  same  form,  inoculation  with  a  dextro-crystal  producing  the 
dextro-form  and  inoculation  with  a  levo-crystal  producing  the 
levo-form. 

Method  of  Formation  of  Derivatives.  In  this  method  an  optic- 
ally active  substance,  generally  an  alkaloid,  is  added  to  the  racemic 
modification,  producing  optically  active  derivatives  having  differ- 


LIQUIDS  109 

ent  solubilities.  Thus  if  cinchonine,  an  optically  active  alkaloid 
having  the  formula,  Ci9H22N20,  be  added  to  the  racemic  modifica- 
tion of  tartaric  acid,  the  cinchonine  salt  of  the  levo-acid  will 
crystallize  first.  The  crystals  of  the  cinchonine  salt  are  then 
removed  and  after  adding  ammonia  to  displace  the  alkaloid, 
dilute  sulphuric  acid  is  added  and  the  pure  levo-tartaric  acid  is 
obtained. 

Method  of  Ferments.  Notwithstanding  the  fact  that  optical 
antipodes  resemble  each  other  so  closely  in  most  of  their  properties, 
Pasteur  found  that  certain  micro-organisms  have  the  power  of 
distinguishing  sharply  between  these  forms.  For  example,  if 
penicillium  glaucum  be  introduced  into  a  solution  of  racemic 
tartaric  acid,  it  thrives  at  the  expense  of  the  dextro-acid  and 
eventually  leaves  the  pure  levo-form.  In  this  method  one  of  the 
active  modifications  is  always  lost.  Pasteur  was  the  first  to 
point  out  that  there  must  be  some  intimate  connection  between 
optical  activity  and  the  constitution  of  the  molecule.  It  remained 
for  Le  Bel  *  and  Van't  Hoff  f  to  formulate  independently  and 
almost  simultaneously  an  hypothesis  to  account  for  optical  activ- 
ity on  the  basis  of  molecular  constitution.  Their  important 
work  laid  the  foundation  of  spatial  chemistry,  commonly  termed 
stereochemistry  (derived  from  the  Greek  o-repeos  =  a  solid). 

Le  Bel  accepted  Pasteur's  view  that  optical  activity  is  depend- 
ent upon  a  condition  of  asymmetry,  but  whether  this  asymmetry 
is  a  property  of  the  crystal  alone  or  whether  it  belongs  to  the 
molecule  of  the  optically  active  substance,  was  the  question  he 
set  himself  to  answer.  He  found,  on  dissolving  certain  optically 
active  crystals  in  an  inactive  solvent,  that  the  optical  activity 
is  imparted  to  the  solution  and  therefore  he  concluded  that  the 
condition  of  asymmetry  must  exist  in  the  chemical  molecule. 
All  of  the  optically  active  substances  known  to  Le  Bel  were 
compounds  of  carbon.  An  examination  of  the  formulas  of  these 
compounds  led  him  to  ascribe  the  cause  of  their  optical  activity 
to  the  presence  of  an  asymmetric  carbon  atom,  that  is  a  carbon 
atom  combined  with  four  different  atoms  or  groups  of  atoms. 

*  Bull.  Soc.  Chim.  (2),  22,  337  (1874). 
t  Ibid.  (2),  23,  295  (1875). 


110 


THEORETICAL  CHEMISTRY 


One  of  the  simplest  examples  is  afforded  by  lactic  acid,  the  struc- 
tural formula  of  which  is 

H 

I 
CH3  -  C  -  COOH 

I 

OH 

In  this  formula  the  asymmetric  carbon  atom  is  placed  at  the 
center  and  is  in  combination  with  hydrogen,  hydroxyl,  methyl 
and  carboxyl.  In  connection  with  his  work  on  the  relation 
between  optical  activity  and  asymmetry,  Le  Bel  pointed  out  that 
active  forms  never  result  from  laboratory  syntheses,  the  racemic 
modification  being  invariably  obtained.  Van't  Hoff  reached  con- 
clusions similar  to  those  of  Le  Bel  and  proposed  the  additional 


HO 


COOH 


COOH 


Fig.  31. 


theory  of  the  asymmetric  tetrahedral  carbon  atom.  Since  the  four 
valences  of  the  carbon  atom  are  equivalent,  as  the  work  of  Henry 
on  methane  has  shown  them  to  be,  Van't  Hoff  pointed  out  that 
the  only  possible  geometrical  arrangement  of  the  atoms  in  the 
molecule  of  methane,  must  be  that  in  which  the  carbon  atom  is 
placed  at  the  center  of  a  regular  tetrahedron  with  the  four 
hydrogen  atoms  at  the  four  apices.  He  then  pointed  out  that 
when  the  four  valences  of  the  tetrahedral  carbon  atom  are  satis- 
fied with  different  atoms  or  groups,  no  plane  of  symmetry  can  be 
passed  through  the  figure,  the  carbon  atom  being  asymmetric. 
This  conception  of  Le  Bel  and  Van't  Hoff  forms  the  basis  of  all 
stereochemistry,  and  has  proved  of  inestimable  value  to  the 
organic  chemist  in  enabling  him  to  explain  the  existence  of  many 
isomeric  compounds.  Thus,  ordinary  lactic  acid  can  be  split  into 


LIQUIDS 


111 


two  optically  active  isomers.  Aside  from  the  fact  that  one  acid 
is  dextro-  and  the  other  is  levo-rotatory,  the  properties  of  the 
two  acids  are  practically  identical.  If  the  formulas  are  written 
spatially,  the  different  groups  can  be  arranged  about  the 
asymmetric  carbon  atom  in  such  a  way  that  the  two  tetrahedra 
shall  be  mirror  images  of  each  other,  as  shown  in  Fig.  31.  It  will 
be  observed  that  these  two  tetrahedra  can  in  no  way  be  super- 
posed so  that  the  same  groups  fall  over  each  other,  that  is  to 
say  they  are  enantiomorphous  orms.  In  tartaric  acid  there  are 
two  asymmetric  carbon  atoms  as  is  evident  when  its  structural 
formula  is  written  as  follows :  — 

H      H 

I  I 
HOOC  -  C  -  C  -  COOH 

I  I 
OH  OH 

If  the  stereochemical  formulas  of  the  dextro-  and  levo-acids  be 
represented  as  in  Fig.  32,  (a)  and  (b),  it  will  be  apparent  that 
the  theory  admits  of  the  existence  of  another  isomer  with  the 
atoms  and  groups  arranged  as  in  Fig.  32  (c). 

Racemic  Acid 
A 


HO 


COOH 
d-Tartaric  Acid 

(a) 


OH 


COOH 
Z-Tartaric  Acid 

ib) 
Fig.  32. 


OH 


COOH 

Meao-Tartaric  Acid 
(0 


In  this  arrangement  the  asymmetry  of  the  upper  tetrahedron 
is  the  reverse  of  that  of  the  lower,  and  consequently  the  optical 
activity  of  one-half  of  the  molecule  exactly  compensates  the  optical 


112  THEORETICAL  CHEMISTRY 

activity  of  the  other  half,  and  the  molecule  as  a  whole  is  inactive. 
It  is  evident  that  such  a  tartaric  acid  could  not  be  split  into  two 
active  forms.  Actually  there  are  four  tartaric  acids  known,  viz., 
(1)  inactive  racemic  acid  which  is  separable  into  (2)  dextro-tartaric 
acid  and  (3)  levo-tartaric  acid;  and  (4)  meso-tartaric  acid,  an  in- 
active substance  which  has  never  been  separated  into  two  active 
forms,  but  which  has  the  same  formula,  the  same  molecular 
weight  and  in  general  the  same  properties  as  the  dextro-  or  levo- 
tartaric  acids.  Inactive  forms,  such  as  meso-tartaric  acid,  are  said 
to  be  inactive  by  internal  compensation.  This  constitutes  one  of 
many  beautiful  confirmations  of  the  Van't  Hoff  theory  of  the 
asymmetric  tetrahedral  carbon  atom. 

Meso-tartaric  acid  furnishes  an  illustration  of  the  fact  that 
asymmetric  carbon  atoms  may  be  present  in  the  molecule  with- 
out imparting  optical  activity  to  the  substance.  The  converse 
of  this  proposition,  however,  that  optical  activity  is  dependent 
upon  asymmetric  carbon  atoms,  is  generally  true.  Quite  recently 
some  substances  apparently  containing  no  asymmetric  carbon 
atoms  have  been  discovered  which  are  optically  active.  An 
example  of  such  a  substance  is  1-methyl  cyclohexylidene-4  acetic 
acid,  to  which  the  following  formula  has  been  assigned:  — 


CH,CH  xC:CH.COOH 

\r<TT     r^TT 

V^X±2  *  v^Jl2 

Other  atoms  aside  from  carbon  may  be  asymmetric;  thus  certain 
compounds  of  nitrogen,  sulphur  and  tin  have  been  shown  to  be 
optically  active.  The  theory  also  furnishes  an  explanation  of 
the  fact,  pointed  out  by  Le  Bel,  that  optically  active  forms  are 
never  obtained  by  direct  synthesis.  Since  the  rotatory  power  is 
dependent  upon  the  arrangement  of  the  atoms  and  groups  in  the 
molecule,  it  follows  from  the  doctrine  of  probability  that  as  many 
dextro  as  levo  configurations  will  be  formed  and  consequently  the 
racemic  modification  will  be  obtained.  Up  to  the  present  time  no 
satisfactory  generalization  has  been  discovered  as  to  the  factors 
determining  the  molecular  rotation  in  any  particular  case.  An 


LIQUIDS  113 

attempt  in  this  direction  has  been  made  by  Guye,*  in  which  he 
ascribes  the  magnitude  of  the  observed  rotation  to  the  relative 
masses  of  the  atoms  or  groups  which  are  in  combination  with  the 
tetrahedral  carbon  atom.  But  it  cannot  be  mass  alone  which 
conditions  optical  activity,  since  substances  are  known  which 
rotate  the  plane  of  polarization  notwithstanding  the  fact  that 
their  molecules  have  two  groups  of  equal  mass  in  combination 
with  the  asymmetric  carbon  atom.  The  molecular  rotations  of 
the  members  of  homologous  series  exhibit  some  regularities,  but 
on  the  other  hand  many  exceptions  occur  which  cannot  be  satis- 
factorily explained.  About  all  that  can  be  said  at  the  present 
time  is,  that  optical  activity  is  a  constitutive  property. 

Magnetic  Rotation.  That  many  substances  acquire  the  power 
of  rotating  the  plane  of  polarized  light  when  placed  in  an  intense 
magnetic  field  was  first  observed  by  Faraday  f  in  1846. 

The  relation  between  chemical  composition  and  magnetic 
rotatory  power  has  since  been  investigated  very  exhaustively  by 
W.  H.  Perkin,{  his  experiments  in  this  field  having  been  continued 
for  more  than  fifteen  years.  In  brief,  Perkin's  method  of  investi- 
gating magnetic  rotatory  power  consisted  in  introducing  the 
liquid  to  be  examined  into  a  polarimeter  tube  1  decimeter  in 
length  and  then  placing  the  tube  axially 
betweeen  the  perforated  poles  of  a 
powerful  electromagnet,  as  shown  in 
Fig.  33.  Upon  exciting  the  magnet  it 
was  found  that  the  plane  of  polarization 
had  been  rotated,  either  to  the  right 
or  the  left,  the  direction  of  rotation 
depending  upon  the  direction  of  the 
current,  the  intensity  of  the  magnetic 

field  and  the  nature  of  the  liquid.  Perkin  used  the  sodium  flame 
as  his  source  of  light  and  carried  out  all  of  his  experiments 
at  15°  C.  He  expressed  his  results  by  means  of  the  formula, 

*  Compt.  rend.,  no,  714  (1890). 

t  Phil.  Trans.,  136,  1  (1846). 

j  Jour,  prakt.  Chem.  [2],  31,  481  (1885);  Jour.  Chem.  Soc.,  49,  777;  41, 
808;  53,  561,  695;  59,  981;  61,  287,  800;  63,  57;  65,  402,  815;  67,  255;  69, 
1025  (1886-1896). 


114  THEORETICAL  CHEMISTRY 

Ma/d,  a  being  the  observed  angle  of  rotation,  d,  the  density 
of  the  liquid  and  M  its  molecular  weight.  All  measurements 
were  expressed  in  terms  of  water  as  a  standard:  thus  if  Ma/d 
is  the  rotation  for  any  substance  and  M'a' /df  is  the  corre- 
sponding rotation  for  water,  then,  according  to  Perkin,  the 
molecular  magnetic  rotation  will  be  given  by  the  ratio,  Ma/d  : 
M'a!  jd'  or  Mad'/Mfa'd. 

The  molecular  magnetic  rotation  for  a  large  number  of  organic 
compounds  has  been  determined  by  Perkin,  who  has  shown  it  to 
be  an  additive  property.  In  any  one  homologous  series  the  value 
of  the  molecular  magnetic  rotation  is  given  by  the  formula 

mol.  mag.  rotation  =  a  +  rib, 

where  a  is  a  constant  characteristic  of  the  series,  b  is  a  constant 
corresponding  to  a  difference  of  CH2  in  composition,  its  value 
being  1.023,  and  n  is  the  number  of  carbon  atoms  contained  in 
the  molecule.  This  formula  is  applicable  only  to  compounds 
which  are  strictly  homologous,  isomeric  substances  in  two  differ- 
ent series  having  quite  different  rotations.  The  constitution  of 
the  molecule  exerts  as  great  an  influence  on  magnetic  rotation 
as  it  does  on  refraction,  a  double  bond  causing  an  appreciable 
increase  in  the  value  of  a.  The  results  of  experiments  on  mag- 
netic rotation  show  that  nothing  like  the  same  regularities  exist 
as  have  been  discovered  for  molecular  refraction  and  molecular 
volume.  The  rotatory  powers  of  various  inorganic  substances 
have  been  determined,  but  the  results  are  too  irregular  to  admit 
of  any  satisfactory  interpretation. 

Absorption  Spectra.  When  a  beam  of  white  light  is  passed 
through  a  colored  liquid  or  solution  and  the  emergent  beam  is 
examined  with  a  spectroscope,  a  continuous  spectrum  crossed  by 
a  number  of  dark  bands  is  obtained.  A  portion  of  the  light  has 
been  absorbed  by  the  liquid.  Such  a  spectrum  is  known  as  an 
absorption  spectrum.  If  instead  of  passing  the  light  through  a 
liquid  it  is  passed  through  an  incandescent  gas,  a  spectrum  will 
be  obtained  which  is  crossed  by  numerous  fine  lines,  termed 
Fraunhofer  lines.  Such  lines  occupy  the  same  positions  as  the 
corresponding  colored  lines  in  the  emission  spectrum  of  the  gas. 


LIQUIDS 


115 


It  follows,  therefore,  that  the  absorption  spectrum  is  quite  as  char- 
acteristic of  a  substance  as  its  emission  spectrum,  and  from  a 
careful  study  of  the  absorption  spectra  of  liquids  we  may  expect 
to  gain  some  insight  into  their  molecular  constitution.  The 
pioneer  workers  in  this  field  were  Hartley  and  Baly  *  and  it  is 
largely  to  them  that  we  owe  our  present  experimental  methods. 
The  instrument  employed  for  photographing  spectra  is  called  a 


Fig.  34. 

spectrograph,  a  very  satisfactory  form  being  shown  in  Fig.  34. 
It  differs  from  an  ordinary  spectroscope  in  that  the  eye-piece  is 
replaced  by  a  photographic  camera.  This  attachment  is  clearly 
shown  in  the  illustration.  The  plateholder  is  so  constructed  that 
only  a  narrow  horizontal  strip  of  the  plate  is  exposed  at  any  one 
time,  thus  making  it  possible  to  take  a  series  of  photographs  on 
the  same  plate  by  simply  lowering  the  holder.  By  means  of  a 
millimeter  scale,  also  shown  in  the  illustration,  the  plateholder 
can  be  moved  through  the  same  distance  each  time  before  expos- 

*  See  numerous  papers  in  the  Jour.  Chem.  Soc.,  since  1880. 


116  THEORETICAL  CHEMISTRY 

ing  a  fresh  portion  of  the  plate,  thus  insuring  an  equally-spaced 
series  of  spectrum  photographs.  In  order  that  spectra  in  the  ultra- 
violet region  may  be  photographed,  it  is  customary  to  equip  the  in- 
strument with  quartz  lenses  and  a  quartz  prism,  ordinary  glass  not 
being  transparent  to  the  ultra-violet  rays.  Using  a  spectrograph 
furnished  with  a  quartz  optical  system,  it  is  possible  to  photograph 
on  a  single  plate  the  entire  spectrum  from  2000  to  8000  Angstrom 
units.  A  scale  of  wave-lengths  photographed  on  glass  is  provided 
with  the  instrument  so  that  the  wave-lengths  of  lines  or  bands 
can  be  read  off  directly  by  laying  the  scale  over  the  photographs. 

The  source  of  light  to  be  used  depends  upon  the  character  of 
the  investigation.  If  a  source  rich  in  ultra-violet  rays  is  desired, 
the  light  from  the  electric  spark  obtained  between  electrodes  pre- 
pared from  an  alloy  of  cadmium,  lead  and  tin  is  very  satisfactory; 
or  the  light  from  an  arc  burning  between  iron  electrodes  may  be 
used.  For  investigations  in  the  visible  region  of  the  spectrum 
the  Nernst  lamp  is  unsurpassed.  In  using  the  spectrograph  for 
the  purpose  of  studying  the  constitution  of  a  dissolved  substance, 
it  is  necessary  to  determine  not  only  the  number  and  position  of 
the  absorption  bands,  but  also  the  persistence  of  these  bands  as 
the  solution  is  diluted. 

According  to  Beer's  law  the  product  of  the  thickness,  t,  of  an 
absorbing  layer  of  solution  of  molecular  concentration,  m,  is  con- 
stant, or  mt  =  k.  If  then  the  thickness  of  a  given  layer  of  solu- 
tion is  diminished  n  times,  its  absorption  will  be  the  same  as  that 
of  a  solution  whose  concentration  is  only  I /nth  of  that  of  the  original 
solution.  Thus,  by  varying  the  thickness  of  the  absorbing  layer 
we  can  produce  the  same  effect  as  by  changing  the  concentration. 
The  convenient  device  of  Baly  for  altering  the  length  of  the 
absorbing  column  of  liquid  is  shown  in  Fig.  35  attached  to  the 
collimator  of  the  spectrograph.  It  consists  of  two  closely-fitting 
tubes,  one  end  of  each  tube  being  closed  by  a  plane,  quartz  disc. 
The  outer  tube  is  fitted  with  a  small  bulbed-f  unnel  and  is  graduated 
in  millimeters.  The  two  tubes  are  joined  by  means  of  a  piece  of 
rubber  tubing  which  prevents  leakage  of  the  contents,  and  at  the 
same  time  admits  of  the  adjustment  of  the  column  of  liquid  to 
the  desired  length  by  simply  sliding  the  smaller  tube  in  or  out. 


LIQUIDS 


117 


Molecular  Vibration  and  Chemical  Constitution.  There  are 
two  systems  of  graphic  representation  of  the  results  of  spectro- 
scopic  investigations.  In  the  first  system,  due  to  Hartley,  the 
wave-lengths  or  their  reciprocals,  the  frequencies,  are  plotted  as 
abscissae  and  the  thicknesses  of  the  absorbing  layers,  in  milli- 
meters, are  plotted  as  ordinates.  Such  curves  are  known  as 
curves  of  molecular  vibration.  The  second  system,  due  to  Baly 
and  Desch,  is  a  modification  of  that  developed  by  Hartley. 


Fig.  35. 

Baly  and  Desch  suggested  that  for  various  reasons  it  would  be 
more  advantageous,  if  instead  of  plotting  the  thicknesses  of  the 
absorbing  layers  as  ordinates,  the  logarithms  of  these  thicknesses 
be  plotted.  Both  methods  have  their  advantages  and  both  are 
used.  As  an  illustration  of  the  value  of  curves  of  molecular 
vibration  in  connection  with  questions  of  chemical  constitution, 
we  will  take  the  case  of  o-hydroxy-carbanil.  The  constitution 
of  this  substance  was  known  to  be  represented  by  one  of  the  two 
following  formulas:  — 

/°\  /°\ 

CeH/      ^>CO,    or    CeH^/        V;  -  OH. 

NH  N7 

(a)  (b) 

Hartley  showed  that  the  absorption  spectra  of  the  two  ethyl 
derivatives,  the  lactam  and  lactim  ethers,   are  very  different. 


118 


THEORETICAL  CHEMISTRY 


On  comparing  the  curves  of  molecular  vibration  for  the  three  sub- 
stances (Fig.  36),  it  is  apparent  that  the  curves  for  the  lactam 
ether  and  o-hydroxy-carbanil  bear  a  close  resemblance  to  each 
other,  while  the  curve  for  the  lactim  ether  is  very  different  from 
the  curves  for  the  other  two  substances.  The  constitution  of 


4000 


i 


J I \ I I L 


Oscillation  Frequencies 

4000 


3456789 


1  2 


-2 


-1 


3456789 


1  2 


456789 


123 


Lactam  Ether 


0-Hydroxy-Cacbanil 
Fig.  36. 


Lactim  Ether 


o-hydroxy-carbanil  must  then  be  very  similar  to  that  of  the 
lactam  ether.  The  formulas  of  the  ethyl  derivatives  of  the 
mother  substance  are  known  to  be  as  follows  :  — 


O 

N-C2H5 
Lactam  ether 


C6H 


-O-C2H6 


Lactim  ether 


LIQUIDS  119 

Hartley  concluded,  therefore,  that  formula  (a)  represents  the 
structure  of  the  molecule  of  o-hydroxy-carbanil. 

It  is  beyond  the  scope  of  this  book  to  discuss  at  greater  length 
the  bearing  of  absorption  spectra  upon  chemical  constitution;  but 
the  student  is  earnestly  advised  to  consult  some  book  *  treating  of 
this  important  subject  or  to  read  some  of  the  original  papers. 

Surface  Tension.  The  attraction  between  the  molecules  of  a 
liquid  manifests  itself  near  the  surface  where  the  molecules  are 
subject  to  an  unbalanced  internal  force.  The  condition  of  a 
liquid  near  its  surface  is  roughly  depicted  in  Fig.  37,  where  the 
dots  A,  B,  and  C  represent  molecules  and  the  circles  represent  the 
spheres  within  which  lie  all  of  the  other  molecules  which  exert  an 
appreciable  attraction  upon  A,  B,  and  C.  The  shaded  portions  rep- 


© 


Fig.  37. 


resent  those  molecules  whose  attractions  are  unbalanced.  These 
unbalanced  forces  will  evidently  tend  to  diminish  the  surface  to 
a  minimum  value,  and  consequently  work  must  be  done  to  increase 
the  surface  of  a  liquid.  The  work  necessary  to  form  a  liquid  sur- 
face one  square  centimeter  in  area  is  termed  the  surface  tension  of  the 
liquid.  Some  liquids  wet  the  walls  of  a  glass  capillary  tube  while 
others  do  not.  When  the  liquid  wets  the  tube,  the  surface  is  con- 
cave and  the  liquid  rises  in  the  tube;  on  the  other  hand  when  the 
liquid  does  not  adhere,  the  surface  is  convex  and  the  liquid  is 
depressed  in  the  tube.  The  law  governing  the  elevation  or  de- 
^pression  of  a  liquid  in  a  capillary  tube  was  discovered  by  Jurin 
and  may  be  stated  thus :  —  The  elevation  or  depression  of  a  liquid 
in  a  capillary  tube  is  inversely  proportional  to  the  diameter  of  the 

*  Relation  between  Chem.  Constitution  and  Phys.  Properties.    Samuel 
Smiles. 


120 


THEORETICAL  CHEMISTRY 


tube.  Let  Fig.  38  represent  a  capillary  tube  of  radius  r,  immersed 
in  a  vessel  of  liquid  whose  density  is  d,  and  let  the  elevation  of 
the  liquid  in  the  capillary  be  denoted  by  h.  Then  the  weight  of 
the  column  of  liquid  in  the  capillary  will  be  Trr2hdg,  where  g  is  the 
acceleration  due  to  gravity.  The  force  sustaining  this  weight  is 


Fig.  38. 

2  irry  cos  6,  the  vertical  component,  of  the  force  due  to  the  tension 
of  the  liquid  surface  at  the  walls  of  the  tube,  7  being  the  surface 
tension  and  6  the  angle  of  contact  of  the  liquid  surface  with  the 
walls  of  the  tube. 
Therefore 

irr2hdg  —  2  irry  cos  0, 
or 

hdgr 


y  = 


2cos0 


In  the  case  of  water  and  many  other  liquids  6  is  so  small  that  we 
may  write  0  =  0,  the  foregoing  expression  becoming 

7  =  1/2  hdgr. 

Thus  the  surface  tension  of  a  liquid  can  be  calculated  provided  its 
density  and  the  height  to  which  it  rises  in  a  previously  calibrated 
tube  is  known.  When  h  and  r  are  expressed  in  centimeters,  7 
will  be  expressed  in  dynes  per  centimeter  or  ergs  per  square  centi- 
meter. A  simple  form  of  apparatus  for  the  determination  of 
surface  tension  used  by  the  author  is  shown  in  Fig.  39.  A  capil- 
lary tube,  A,  of  uniform  bore  is  sealed  to  a  glass  rod,  E,  which  is 
held  in  position  in  the  test  tube,  B,  by  means  of  a  cork  stopper. 
A  short  right-angled  tube,  Z>,  and  a  thermometer,  F,  are  also 


LIQUIDS 


121 


passed  through  the  same  cork  stopper.  The  liquid  whose  surface 
tension  is  to  be  measured  is  introduced  into  the  tube,  B,  the  cork 
inserted  and  the  tube  placed  inside 
of  the  larger  tube,  C,  containing  a 
liquid  of  known  boiling-point. 
When  the  thermometer,  F,  has 
become  stationary,  the  capillary 
elevation  of  the  liquid  is  measured  G 
with  a  cathetometer.  The  tube, 
Z),  permits  the  escape  of  vapor 
from  the  liquid  in  B  and  at  the 
same  time  insures  equality  of  pres- 
sure inside  and  outside  of  the  ap- 
paratus. The  spiral  tube,  G,  serves 
as  an  air  condenser,  preventing 
loss  of  vapor  from  the  liquid  in  the 
outer  tube.  The  surface  tension 
of  a  liquid  has  been  found  to  depend 
upon  the  nature  of  the  liquid  and 
also  upon  its  temperature. 

Surface  Tension  and  Molecular 
Weight.  In  1886,  Eotvos  *  showed 
that  the  surface  tension  multiplied 
by  the  two-thirds  power  of  the  mo- 
lecular weight  and  specific  volume 
is  a  function  of  the  absolute  tem- 
perature, or 

7  (Mv)l  =  f(T), 

Fig.  39. 
where  7  is  the  surface  tension,  M 

the  molecular  weight,  v  the  specific  volume  or  reciprocal  of  the 
density,  and  T  the  absolute  temperature.  Ramsay  and  Shields  t 
modified  the  equation  of  Eotvos  as  follows:  — 

7  (Mv)l  =  k(tc-t-  6),  (1) 

tc  being  the  critical  temperature  of  the  liquid,  t  the  temperature 
of  the  experiment,  and  k  a  constant  independent  of  the  nature  of 

*  Wied.  Ann.,  27,  448  (1886). 

t  Zeit.  phys.  Chem.,  12,  431  (1893). 


122  THEORETICAL  CHEMISTRY 

the  liquid.  The  physical  significance  of  the  two-thirds  power  of 
the  molecular  volume  has  been  explained  by  Ostwald  in  the  follow- 
ing manner  :  —  Assuming  the  molecules  to  be  spherical,  we_shall 
have  for  two  different  liquids,  the  proportion 

Fi:Fa::ri8:r*8, 

where  Fi  and  F2  represent  the  volumes  and  ri  and  r2  the  radii  of 
their  respective  molecules.  Similarly  the  ratio  of  the  surfaces, 
Si  and  S2,  of  the  molecules  in  terms  of  their  respective  radii,  will 
be 

Si:&::ria:r2». 

From  these  two  proportions  it  follows  that  the  ratio  of  the  molec- 
ular surfaces  in  terms  of  the  molecular  volumes,  will  be 

8i:&::Vi*:Vj. 

Making  use  of  the  value  of  M  as  determined  in  the  gaseous  state, 
Ramsay  and  Shields  found  the  value  of  k  for  a  large  number  of 
liquids  to  be  equal  to  2.12  ergs.  Among  the  liquids  for  which 
this  value  of  k  was  found  were  benzene,  carbon  tetrachloride, 
carbon  disulphide  and  phosphorus  trichloride.  For  certain  other 
liquids  such  as  water,  methyl  and  ethyl  alcohols  and  acetic  acid, 
k  was  found  to  have  values  much  smaller  than  2.12.  Ramsay 
and  Shields  attributed  these  abnormalities  to  an  increase  in  molec- 
ular weight  due  to  association,  and  suggested  that  the  degree  of 
association  might  be  calculated  from  the  equation 

a*  =  2.12/fc', 
or 


/2. 

HIT 


(2) 


where  x  denotes  the  factor  of  association,  and  k'  is  the  value  of 
the  constant  for  the  associated  liquid  in  equation  (1).  It  was 
further  pointed  out  by  Ramsay  and  Shields  that  equation  (1) 
affords  a  means  of  calculating  the  molecular  weight  of  a  pure 
liquid,  provided  we  assume  that  for  a  non-associated  liquid  the 
mean  value  of  k  is  2.12.  Since  it  is  not  an  easy  matter  to  deter- 
mine the  critical  temperature  with  accuracy,  Ramsay  and  Shields 
made  use  of  a  differential  method,  and  thus  eliminated  tc  from 
equation  (1).  If  the  surface  tension  of  a  liquid  be  measured  at 


LIQUIDS  123 

two  temperatures  ^and  k,  and  the  corresponding  densities  are  di 
and  d2,  we  shall  have 

71  (M/di)l  =  k(tc-tl-  6),  (3) 
and 

72  (M/di)l  =  k(tc-k-  6).  (4) 
Subtracting  equation  (4)  from  equation  (3),  we  obtain 

71  (M/di)l  -  72 


or  solving  equation  (5)  for  M,  we  have 


V  7i-T2 

The  method  of  Ramsay  and  Shields  is  the  best  known  method  for 
the  determination  of  the  molecular  weight  of  a  pure  liquid.  If 
M  is  known  to  be  the  same  in  the  liquid  and  gaseous  states, 
or"  in  other  words,  if  k  is  independent  of  the  temperature,  even 
though  its  value  is  not  exactly  2.12,  the  critical  temperature  of 
the  liquid  can  be  calculated  by  means  of  equation  (1).  In  order 
that  the  correct  value  of  the  critical  temperature  may  be  obtained, 
Ramsay  and  Shields  found  it  necessary  to  use  the  specific  value 
of  k  for  the  liquid  whose  critical  temperature  is  sought.  As  an 
illustration  of  the  method  of  calculation,  the  following  example  is 
taken  from  the  work  of  Ramsay  and  Shields. 
For  carbon  disulphide, 

7  at  19°.4  =  33.58  7  at  46°.l  =  29.41 

d  at  19°.4  =  1.264  d  at  46°.l  =  1.223. 

We  have  then  for  7  (M/d)l,  at  the  two  temperatures, 

(76/1.264)1  X  33.58  =  515.4, 
and 

(76/1.223)1  X  29.41  =  461.4. 
Substituting  in  the  equation 

71  (M/dtf  -  72 


-k, 

we  have, 

515.4-461.4 
46.1-19.4    -2'022' 


124  THEORETICAL  CHEMISTRY 

This  value  of  k  is  so  nearly  equal  to  the  mean  value,  2.12,  that  we 
assume  M  to  be  the  same  in  the  liquid  and  gaseous  states,  and 
therefore  we  may  substitute  in  equation  (1)  and  calculate  the 
critical  temperature  of  carbon  disulphide  thus, 

7  (M/d)\  =  k(tc-t-  6), 
or  solving  for  tc,  we  have 


c  . 

Substituting  the  data  given  above,  in  the  preceding  equation,  we 
obtain 

tc  =  515.4/2.022  +  6  +  19.4, 
or  tc  =  280°.3  C. 

Surface  Tension  and  Drop-Weight.  Morgan  and  his  co- 
workers,*  from  measurements  of  the  volumes  of  a  single  drop 
falling  from  the  carefully-ground  tip  of  a  capillary  tube,  have 
shown  that  the  weight  of  the  falling  drop,  from  such  a  tip  can  be 
used  in  place  of  the  surface  tension  in  the  equation  of  Ramsay 
and  Shields  for  the  calculation  of  molecular  weights  and  critical 
temperatures.  The  modified  equation  may  be  written  thus  :  — 

__  y, 


where  w\  amd  wz  are  the  respective  weights  of  the  falling  drop 
at  the  temperatures,  ti  and  t%.  The  value  of  k  obviously  depends 
upon  the  tip  employed. 

The  results  obtained  by  the  drop-weight  method  have  been 
shown  to  be  more  trustworthy  than  those  obtained  by  the  method 
of  capillary  elevation.  Morgan  has  further  pointed  out  that 
when  the  experimental  data  are  substituted  in  the  preceding 
formula,  the  magnification  of  the  experimental  errors  is  appreci- 
ably greater  than  when  use  is  made  of  the  original  formula, 
w  (M/d)\  =  k(tc-t-  6). 

Morgan  recommends  therefore  that  this  formula  be  used  for 
the  determination  of  molecular  weights.     After  having  calibrated 

*  Jour.  Am.  Chem.  Soc.,  30,  360  (1908);  30,  1055  (1908). 


LIQUIDS  125 

a  particular  tip  with  pure  benzene  (a  liquid  which  is  known  to  be 
non-associated),  and  thus  ascertaining  the  value  of  /b,  the  drop- 
weights  at  several  different  temperatures  are  determined.  If 
we  assume  M  to  have  the  same  value  in  the  liquid  and  gaseous 
states,  the  value  of  tc  can  be  computed  by  substituting  the  experi- 
mental data  in  the  preceding  equation.  If  at  the  different  tem- 
peratures at  which  drop-weights  are  determined,  the  same  value 
of  tc  is  obtained,  then  we  may  infer  that  the  liquid  is  non-asso- 
ciated and,  therefore,  that  the  assumption  made  as  to  the  value  of 
M  is  confirmed.  It  is  a  singular  fact  that  the  calculated  value 
of  tc  for  some  liquids  does  not  agree  with  the  experimental  value, 
although  it  remains  constant  throughout  an  extended  range  of 
temperatures.  Morgan  considers  a  constant  value  of  tc  to  be  an 
indication  of  non-association,  even  if  the  value  is  fictitious.  In 
this  method  the  constancy  of  the  calculated  value  of  the  critical 
temperature  becomes  the  criterion  of  molecular  association,  and 
thus  affords  a  means  of  determining  whether  the  molecular  weight 
in  the  liquid  state  is  identical  with  that  in  the  gaseous  state.  The 
values  of  tc  calculated  from  the  drop-weights  of  an  associated 
liquid  become  steadily  smaller  as  the  temperature  increases.  A 
large  number  of  liquids  have  been  studied  by  this  method,  and 
the  results  indicate  that  many  of  the  substances  which  were  con- 
sidered to  be  associated  by  Ramsay  and  Shields  are  in  reality 
non-associated  ;  in  fact,  it  appears  from  the  work  of  Morgan  that 
association  is  much  less  common  among  liquids  than  has  hitherto 
been  supposed. 

Dielectric  Constants.  In  1837,  Faraday  discovered  that  the 
attraction  or  repulsion  between  two  electric  charges  varies  with 
the  nature  of  the  intervening  medium  or  dielectric.  If  q\  and  <?2 
represent  two  charges  which  are  separated  by  a  distance  r,  the 
force  of  attraction  or  repulsion,  /,  is  given  by  the  equation 


where  D  is  a  specific  property  of  the  medium  known  as  the  dielectric 
constant.  The  dielectric  constant  of  air  is  taken  as  unity.  Var- 
ious methods  have  been  devised  for  the  experimental  determination 


126 


THEORETICAL  CHEMISTRY 


of  the  dielectric  constant,  but  the  scope  of  this  book  forbids 
even  a  brief  description  of  the  apparatus  or  an  outline  of  the 
processes  of  measurement.  For  a  description  of  these  methods 
the  student  is  referred  to  any  one  of  the  more  complete  physico- 
chemical  laboratory  manuals,  or_to  the  original  •  communications 
of  Nernst  *  and  Drude.f 

The  values  of  the  dielectric  constants  for  some  of  the  more 
common  solvents  are  given  in  the  accompanying  table. 

DIELECTRIC  CONSTANTS  AT  18°  C. 


Substance. 

D 

Hydrogen  dioxide  

92.8 

Water 

77  0 

Formic  acid 

63  0 

Methyl  alcohol 

33  7 

Ethyl  alcohol 

25  9 

Ammonia,  liquid 

22  0 

Chloroform 

5  0 

Ether  

4  4 

Carbon  disulphide  

2.6 

Benzene 

2  3 

The  importance  of  this  property  of  liquids  will  become  more 
apparent  in  subsequent  chapters,  especially  in  those  devoted  to 
electrochemistry. 


PROBLEMS. 

1.  It  is  desired  to  compare  the  molecular  volumes  of  alcohol  and  ether. 
If  the  molecular  volume  of  ether  is  determined  at  20°  C.,  at  what  temper- 
ature must  the  molecular  volume  of  alcohol  be  determined?  The  boiling 
points  of  alcohol  and  ether  are  78°  and  35°  respectively.      Ans.   51°  C. 

2.  A  volume  of  50  liters  of  air  in  passing  through  a  liquid  at  22°  C. 
causes  the  evaporation  of  5  grams  of  substance,  the  molecular  weight  of 
which  is  100.    What  is  the  vapor  pressure  of  the  liquid  in  grams  per 
square  centimeter?  Ans.  25. 

*  Zeit.  phys.  Chem.,  14,  622  (1894). 
t  Ibid.,  23,  267  (1897). 


LIQUIDS  127 

3.  The  boiling-point  of  ethyl  propionate  is  98°.7  C.  and  its  heat  of 
vaporization  is  77.1  calories.     Calculate  its  molecular  weight. 

4.  The  heat  of  vaporization  of  liquid   ammonia  at  its  boiling-point, 
under  atmospheric  pressure  (—  33°. 5  C.)  is  341  calories.     Is  liquid  am- 
monia associated? 

5.  Calculate  the  molecular  volume  of  ethyl  butyrate.    The  molecular 
volume  determined  by  experiment  is  149.1. 

6.  For  propionic  acid,  d  =  1.0158  and  no  =  1.3953.     Calculate  the 
molecular  refraction  by  the  formula  of  Lorenz-Lorentz  and  compare  the 
value  so  obtained  with  that  derived  from  the  atomic  refractions  of 
the  constituent  elements. 

7.  The  density  of  ether  is  0.7208,  of  ethyl  alcohol,  0.7935  and  of  a 
mixture  of  ether  and  alcohol  containing  p  per  cent  of  the  latter,  0.7389. 
At  20°  C.  the  refractive  indices  for  sodium  light  are,  for  ether,  1.3536,  for 
alcohol,  1.3619,  and  for  the  mixture,  1.3572.     Calculate  the  value  of  p, 
using  the  Gladstone  and  Dale  formula.  Am.  20.81. 

8.  At  20°  C.  the  density  of  chloroform  is  1.4823  and  the  refractive 
index  for  the  D-line  is  1.4472.     Given  the  atomic  refractivities  of  carbon 
and   hydrogen,   calculate   that   of  chlorine,   using   the  Lorenz-Lorentz 
formula.  Ans.   5.999. 

9.  Calculate  the  surface  tension  of  benzene  in  dynes  per  centimete^ 
the  radius  of  the  capillary  tube  being  0.01843  cm.,  the  density  of  the 
liquid,  0.85,  and  the  height  to  which  it  rises  in  the  capillary,  3.213  cm. 

Ans.  24.71  dynes/cm. 

10.  Find  the  molecular  weight  of  benzene,  the  surface  tension  at 
46°  C.  being  24.71  dynes  per  centimeter,  its  critical  temperature,  228°.5  C., 
its  density,  0.85  and  the  value  of  k  =  2.12.  Ans.  78. 

11.  At  14°.8  C.  acetyl  chloride  (density  =  1.124)  ascends  to  a  height 
of  3.28  cm.  in  a  capillary  tube  the  radius  of  which  is  0.01425  cm.    At 
46°.2  C.  in  the  same  tube  the  elevation  is  2.85  cm.  and  the  density 
=  1.064.     Calculate  the  critical  temperature  of  acetyl  chloride. 

Ans.  235°.2C. 

12.  From  a  certain  tip  the  weights  of  a  falling  drop  of  benzene  are 
35.329  milligrams  (temp.  =  11°.4,  density  =  0.888)  and    26.530    milli- 
grams (temp.  =  68°. 5,  density  =  0.827).    The  molecular  weight  is  the 
same  in  the  liquid  and  gaseous  states.     Calculate  the  critical  temper- 
ature of  benzene.  Ans.  288°.4  C. 


CHAPTER  VI. 
SOLIDS. 

General  Properties  of  Solids.  Solids  differ  from  gases  and 
liquids  in  possessing  definite,  individual  forms.  Matter  in  the 
solid  state  is  capable  of  resisting  considerable  shearing  and  tensile 
stresses.  In  terms  of  the  kinetic  theory  of  matter,  the  mutual 
attractive  forces  exerted  by  the  molecules  of  solids  must  be  re- 
garded as  superior  to  the  attractive  forces  between  the  molecules 
of  gases  and  liquids.  With  one  or  two  exceptions  all  solids  ex- 
pand when  heated,  but  there  is  no  simple  law  expressing  the  relation 
between  the  increment  of  volume  and  the  temperature.  Rigidity  is 
another  characteristic  property  of  solids,  it  being  much  more  ap- 
parent in  some  than  in  others.  Many  solids  are  constantly  under- 
going a  process  of  transformation  into  the  gaseous  state  at  their 
free  surfaces,  such  a  change  being  known  as  sublimation.  Just 
as  when  a  gas  is  sufficiently  cooled  it  passes  into  the  liquid  state, 
so  on  cooling  a  liquid  below  a  certain  temperature,  it  passes  into 
the  solid  state.  The  reverse  transformations  are  also  possible,  a 
solid  being  liquefied  when  sufficiently  heated,  and  the  resulting 
liquid  completely  vaporized  if  the  heating  be  continued.  Heat 
energy  is  required  to  effect  transition  from  the  solid  to  the  liquid 
state,  just  as  heat  energy  is  required  to  effect  transition  from  the 
liquid  to  the  gaseous  state. 

Obviously  a  substance  in  the  solid  state  contains  less  energy 
than  it  does  in  the  liquid  state.  The  number  of  calories  required 
to  melt  1  gram  of  a  solid  substance  is  called  its  heat  of  fusion. 
It  is  often  difficult  to  decide  whether  a  substance  should  be  classi- 
fied as  a  solid  or  as  a  liquid.  For  example  the  behavior  of  certain 
amorphous  substances  such  as  pitch,  amber  and  glass,  is  similar 
to  that  of  a  very  viscous,  inelastic  liquid.  Solids  are  generally 
classified  as  crystalline  or  amorphous.  In  crystalline  solids  the 
molecules  are  supposed  to  be  arranged  in  some  definite  order,  this 

128 


SOLIDS  129 

arrangement  manifesting  itself  in  the  crystal  form.  An  amorphous 
solid  on  the  other  hand  may  be  considered  as  a  liquid  possessing 
great  viscosity  and  small  elasticity.  The  physical  properties  of 
amorphous  solids  have  the  same  values  in  all  directions,  whereas  in 
crystalline  solids  the  values  of  these  properties  are  different  in 
different  directions.  When  an  amorphous  solid  is  heated  it 
gradually  softens  and  eventually  acquires  the  properties  charac- 
teristic of  a  liquid,  but  during  the  process  of  heating  there  is  no 
definite  point  of  transition  from  the  solid  to  the  liquid  state.  On 
the  other  hand  when  a  crystalline  solid  is  heated  there  is  a  sharp 
change  from  one  state  to  the  other  at  a  definite  temperature, 
this  temperature  being  termed  the  melting-point. 

Crystallography.  The  study  of  the  definite  geometrical  forms 
assumed  by  cyrstalline  solids  is  termed  crystallography.  The 
number  of  crystalline  forms  known  is  exceedingly  large,  but  it  is 
possible  to  reduce  the  many  varieties  to  a  few  classes  or  systems 
by  referring  their  principle  elements  —  the  planes  —  to  definite 
lines  called  axes.  These  axes  are  so  drawn  within  the  crystal  that 
the  crystal  surfaces  are  symmetrically  arranged  about  them. 
This  system  of  classification  was  proposed  by  Weiss  in  1809. 
He  showed  that  notwithstanding  the  multiplicity  of  crystal  forms 
encountered  in  nature,  it  is  possible  to  consider  them  as  belonging 
to  one  or  the  other  of  six  systems  of  crystallization. 

The  six  systems  of  Weiss  are  as  follows:  — 

1.  The  Regular  System.     Three  axes   of   equal    length,  inter- 
secting each  other  at  right  angles,  (Fig.  40  a). 

2.  The  Tetragonal  System.     Two  axes  of  equal  length  and  the 
third  axis  either  longer  or  shorter,  all  three  axes  intersecting  at 
right  angles,  (Fig.  41  a). 

3.  The  Hexagonal  System.     Three  axes  of  equal  length,  all  in 
the  same  plane  and  intersecting  at  angles  of  60°,  and  a  fourth  axis, 
either  longer  or  shorter  and  perpendicular  to  the  plane  of  the 
other  three  (Fig.  42  a). 

4.  The  Rhombic  System.     Three  axes  of  unequal  length,  all 
intersecting  each  other  at  right  angles  (Fig.  43  a). 

5.  The  Monoclinic  System.     Three  axes  of  unequal  length,  two 


130 


THEORETICAL  CHEMISTRY 


(a) 


(0) 


(6) 


coOo 


(O 


SOLIDS  131 

of  which  intersect  at  right  angles,  while  the  third  axis  is  per- 
pendicular to  one  and  not  to  the  other  (Fig.  44  a). 

6.  The  Triclinic  System.  Three  axes  of  unequal  length  no  two 
of  which  intersect  at  right  angles  (Fig.  45  a). 

The  position  of  a  plane  in  space  is  determined  by  three  points 
in  a  system  of  coordinates,  and  consequently  the  position  of  the 
face  of  a  crystal  is  likewise  determined  by  its  points  of  inter- 
section with  the  three  axes,  or,  by  the  distances  from  the  origin 
of  the  system  of  coordinates  at  which  the  plane  of  the  crystal 
face  intersects  the  three  axes.  These  distances  are  called  the 
parameters  of  the  plane.  In  the  regular  system  where  the  three 
axes  are  of  equal  length,  they  are  designated  by  the  same  letter, 
a  in  Fig.  40  a. 

In  the  other  systems  where  the  axes  are  unequal  they  are 
designated  by  a,  b  and  c  (Figs.  41  a-45  a). 

In  the  regular  or  isometric  system,  if  a  plane  intersects  all 
three  axes  at  equal  distances,  as  in  the  regular  octahedron  (Fig. 
40  b),  the  parameter  ratio  is  a  :  a  :  a.  Since  the  value  of  the  in- 
tercepts which  determine  the  position  of  a  given  plane  is  a  re- 
lative value,  one  parameter  is  assumed  as  unity:  if  a  plane 
intersects  the  axes  at  unequal  distances,  the  parameter  ratio  is 
a  :  ma  :  na.  If  the  plane  of  a  crystal  face  be  parallel  to  an  axis 
its  parameter  with  reference  to  that  axis  will  be  infinity,  and 
the  ratio  becomes  a  :  a  :  oo  a.  The  above  notation  due  to  Weiss 
has  been  shortened  by  Naumann,  whose  notation  is  used  in  the 
accompanying  illustrations.  According  to  Naumann  the  ratio 
a  :  a  :  ooa  becomes  ooO  (Fig.  40  d).  The  capital  letter  0  (octa- 
hedron) is  used  when  the  planes  are  referred  to  a  set  of  equal 
axes,  and  P  (pyramid)  (Fig.  41  b)  when  they  are  referred  to 
systems  of  unequal  axes.  The  parameter  before  the  capital 
letter  refers  to  the  vertical  axis,  while  that  after  the  capital 
refers  to  one  of  the  lateral  axes.  Perhaps  the  most  widely  used 
system  of  notation  is  that  due  to  Miller.  In  this  system  the 
reciprocals  of  the  parameters  are  used;  e.g.,  the  ratio  a  :  a  :  ooa 
is  expressed  thus,  110.  A  few  simple  forms  belonging  to  the 
regular  system  together  with  their  parameter  ratios  are  given 
in  Fig.  40  b-j  inclusive. 


132  THEORETICAL  CHEMISTRY 

In  general  the  forms  in  this  system  and  in  all  succeeding  sys- 
tems may  be  grouped  in  the  following  manner: —  (1)  pyram- 
idal forms  which  cut  all  the  axes;  (2)  prismatic  forms  cutting 
two  of  the  axes  and  parallel  to  the  vertical  axis;  and  (3)  pinacoidal 
forms  parallel  to  two  of  the  axes  and  cutting  one  of  the  lateral 
axes.  In  addition  to  these  simple  forms  containing  the  full 
number  of  faces,  the  so-called  holohedral  forms,  other  forms  are 
found  to  exist  which  show  only  one-half,  one-fourth  or  one-eighth 
of  the  number  of  faces  belonging  to  the  perfect  form.  These 
are  called  hemihedral,  tetartohedral  and  ogdohedral  forms  respec- 
tively. These  forms  may  be  imagined  to  result  from  the  en- 
largement of  symmetrically  distributed  faces  of  holohedral  forms 
at  the  expense  of  corresponding  planes  which  are  suppressed. 
Hemihedral,  tetartohedral  and  ogdohedral  faces  are  designated 
by  the  parameter  ratio  of  the  corresponding  holohedral  faces 
divided  by  2,  4  or  8.  Thus  the  tetrahedron,  resulting  from  the 
octahedron  by  the  suppression  of  every  alternate  face,  has  the 
parameter  ratio  0/2,  (Fig.  40k).  Similarly  the  pentagonal 
dodecahedron  (Fig.  40 1)  has  the  parameter  ratio,  m  0  oo  /2,  it 
being  the  hemihedral  form  of  the  tetrahexahedron. 

In  the  tetragonal  system  the  axis  ratio  is  a  :  a  :  c,  the  ratio 
a  :  c  being  definite  but  not  rational.  The  fundamental  form 
has  been  chosen  as  the  pyramid  with  the  ratio  a  :  a  :  c,  or  P 
in  the  Naumann  notation.  If  c  is  less  than  a  then  the  pyramid 
is  acute.  From  these  pyramids  of  the  first  order,  in  which  the 
axes  pass  through  the  angles,  we  must  distinguish  pyramids  of 
the  second  order,  in  which  the  two  secondary  or  lateral  axes 
pass  through  the  middle  points  of  the  edges.  The  planes  of  the 
latter  are  parallel  to  one  secondary  axis,  and  consequently  its 
symbol  must  be  a  :  oo  a  :  c  or  P  oo .  A  combination  of  a  pyra- 
mid of  the  first  with  a  pyramid  of  the  second  order  is  shown 
in  Fig.  41  c.  If  the  planes  of  the  faces  of  the  pyramid  intersect 
the  principal  axis  at  infinity,  the  resulting  form  will  be  the 
quadratic  prism.  A  combination  of  the  quadratic  pyramid  and 
quadratic  prism  is  shown  in  Fig.  41  d. 

All  of  the  forms  mentioned  so  far  fall  into  one  of  three  groups 
of  pyramids,  prisms  or  pinacoids,  but  there  is  still  another  type 


SOLIDS 


133 


which  is  important  in  the  tetragonal  and  succeeding  systems; 
this  is  the  basal  pinacoid,  designated  by  the  symbol  OP.  It 
can  occur  only  in  combination  with  other  forms  and  differs  from 


(a) 


all  other  pinacoids  in  that  it  cuts  the  vertical  axis  and  is  parallel 
to  the  lateral  axes.  In  the  hexagonal  system  the  axis  ratio  is 
a  :  a  :  (a)  :  c,  the  ratio  a  :  c  being  definite  but  not  rational.  The 
fundamental  form  of  this  system  is  the  hexagonal  pyramid,  the 
axis  ratio  of  which  is  a  :  a  :  (oo  a)  :  c  or  P,  from  which  the  hexa- 


(a) 


gonal  prism,  a  :  a  :  (oo  a)  :  oo  c  or  oo  P,  may  be  derived.  In  this 
system  also  occur  pyramids  and  prisms  of  the  first  and  second 
orders,  di-hexagonal  pryamids  and  prisms  and  the  basal  pina- 
coid. One  of  the  common  hemihedral  forms  is  shown  in  Fig. 
42  c  ;  this  is  the  rhombohedron  produced  from  a  pyramid  mP 
by  the  development  of  alternate  faces.  Both  hemihedrism  and 


134 


THEORETICAL  CHEMISTRY 


tetartohedrism   attain  maximum   importance  in  the  hexagonal 
system. 

The  rhombic  system  is  characterized  by  three  unequal  rectan- 
gular axes,  a,  6,  and  c  (Fig.  43  a).  Any  one  axis  as  c  may  be 
selected  as  the  principal  axis,  the  shorter  axis,  a,  being  termed 


0 

(a) 


the  brachy-diagonal  while  the  longer  axis,  b,  is  known  as  the  mac- 
ro-diagonal. In  addition  to  the  unit  pyramid  (Fig.  43  b), 
a  :  b  :  c  or  P,  and  the  unit  prism,  a  :  b  :  oo  c  or  oo  P,  there  are 
the  brachydome,  Poo ,  and  the  macrodome,  Pv5 ,  brachy-  and  macro- 


(a) 


Fig.  44. 


pryamids,  brachy-  and  macro-prisms,  and  the  brachy-  and  macro- 
pinacoids. 

In  the  monoclinic  system  the  axis  b  is  called  the  ortho-diagonal 
while  the  oblique  axis  a  is  called  the  clino-diagonal. 

The  fundamental  form  of  this  system,  the  pyramid,  is  shown 
in  Fig.  44  b  and  is  designated  by  the  symbols  +  P  or  —  P  ac- 


SOLIDS  135 

cording  to  the  position  of  the  planes  in  the  acute  or  obtuse  octant. 
The  terms  ortho-  and  clino-  are  used  in  this  system  to  designate 
planes  in  relation  to  the  ortho-  and  clino-  diagonals. 

The  axis  ratio  for  the  triclinic  system  is  a  :  b  :  c,  the  three 
angles  being  definite  for  each  substance.  The  forms  belonging 
to  this  system  are  necessarily  complicated.  No  form  occurs 
except  in  combination  with  others;  the  triclinic  pyramid  P, 
for  example,  being  made  up  of  four  quarter  pyramids,  Pf  for 
the  upper  right-hand  octant,  and  'P,  Pn  and  f  for  the  other 
three  octants.  The  triclinic  prism  ooP,  (Fig.  45  b),  is  to  be 
regarded  as  made  up  of  two  hemi-prisms,  ooP'  (right-handed) 


Fig.  45. 

and  'ooP  (left-handed).     As  in  the  rhombic  system  the  planes  are 
referred  to  the  brachy-  and  macro-diagonals. 

The  fundamental  law  of  crystallography  discovered  by  Steno 
in  1669,  may  be  stated  thus:  —  The  angle  between  two  given  crystal 
faces  is  always  the  same  for  the  same  substance.  The  fact  that 
every  crystalline  substance  is  characterized  by  a  constant  inter- 
facial  angle,  affords  a  valuable  means  of  identification  which  is 
used  by  both  chemists  and  mineralogists.  The  instrument  em- 
ployed for  the  measurement  of  the  interfacial  angles  of  crystals  is 
called  a  goniometer.  The  crystal  to  be  measured  is  mounted  at 
the  center  of  the  graduated  circular  table  of  the  goniometer,  and 
the  image  of  an  illuminated  slit,  reflected  from  one  surface  of  the 
crystal,  is  brought  into  coincidence  with  the  cross-wires  in  the  eye- 
piece of  the  telescope.  The  table  is  then  turned  until  the  image 


136  THEORETICAL  CHEMISTRY 

of  the  slit,  reflected  from  the  adjacent  face  of  the  crystal,  coincides 
with  the  cross-wires.  The  interfacial  angle  of  the  crystal  is  de- 
termined by  the  number  of  degrees  through  which  the  table  has 
been  turned. 

Properties  of  Crystals.  The  properties  of  all  crystals,  except 
those  belonging  to  the  regular  system,  exhibit  differences  depen- 
dent upon  the  direction  in  which  the  particular  measurements 
are  made.  Thus  the  elasticity,  the  thermal  and  electrical  con- 
ductivities, and  in  fact  all  of  the  physical  properties  of  crystals 
which  do  not  belong  to  the  regular  system,  have  different  values 
in  different  directions.  Crystals  whose  physical  properties  have 
the  same  .values  in  all  directions  are  termed  isotropic,  while  those 
in  which  the  values  are  dependent  upon  the  direction  in  which 
the  measurements  are  made,  are  called  anisotropic.  Crystals 
belonging  to  the  regular  system,  and  amorphous  substances  are 
isotropic.  Certain  amorphous  substances,  such  as  glass,  which 
are  normally  isotropic,  may  become  anisotropic  when  subjected 
to  tension  or  compression.  The  phenomenon  of  double  refrac- 
tion observed  in  all  crystals,  except  those  belonging  to  the  regular 
system,  is  due  to  their  anisotropic  character.  Crystals  belonging 
to  the  tetragonal  and  hexagonal  systems  resemble  each  other  in 
this  respect,  that  in  all  of  them  there  is  one  direction,  called  the 
optic  axis,  or  axis  of  double  refraction  (coincident  with  the  principal 
crystallographic  axis),  along  which  a  ray  of  light  is  singly  re- 
fracted, while  in  all  other  directions  it  is  doubly  refracted.  In 
crystals  belonging  to  the  rhombic,  monoclinic,  and  triclinic  systems, 
there  are  always  two  directions  along  which  a  ray  of  light  is  singly 
refracted.  A  crystal  of  Iceland  spar  (CaCO3)  affords  a  beauti- 
ful illustration  of  double  refraction.  On  placing  a  rhomb  of  this 
substance  over  a  piece  of  white  paper  on  which  there  is  an  ink 
spot,  two  spots  are  seen.  On  turning  the  crystal,  one  spot  will 
remain  stationary  while  the  other  spot  will  revolve  about  it.  This 
property  of  Iceland  spar  is  utilized  in  the  construction  of  Nicol 
prisms  for  polariscopes. 

The  examination  of  sections  of  anisotropic  crystals  in  a  polari- 
scope  between  crossed  Nicol  prisms,  reveals  something  as  to  their 
crystal  form.  As  has  been  stated,  crystals  of  the  tetragonal  and 


SOLIDS  137 

hexagonal  systems  are  uniaxial.  If  a  section  is  cut  from  such  a 
crystal  perpendicular  to  the  optic  axis,  and  this  is  placed  between 
the  crossed  Nicol  prisms  of  a  polariscope,  in  a  convergent  beam 
of  white  light,  a  dark  cross  and  concentric,  spectral-colored  circles 
will  be  observed,  Fig.  46.  Upon  turning  the  analyzer  through 
90°  the  colors  of  the  circles  will  change  to  the  respective  comple- 
mentary colors  and  the  dark  cross  will  become  light.  Crystals 
of  the  rhombic,  monoclinic  and  triclinic  systems  are  biaxial.  If 
a  section  of  a  biaxial  crystal,  cut  perpendicular  to  the  line  bisect- 
ing the  angle  between  the  two  axes,  be  placed  in  the  polariscope 
and  examined  as  in  the  preceding  case,  a  series  of  concentric 
spectral-colored  lemniscates  surrounding  two  dark  centers  and 
pierced  by  dark,  hyperbolic  brushes,  will  be  observed,  as  shown  in 
Fig.  47.  On  rotating  the  analyzer  the  colors  will  change  to  the 


Fig.  46.  Fig.  47. 

corresponding  complementary  colors,  as  in  the  case  of  uniaxial 
crystals.  The  appearance  of  these  figures  is  so  varied  and 
characteristic  as  to  furnish,  in  many  cases,  a  very  satisfactory 
means  of  identifying  anisotropic  crystals. 

Etch  Figures.  The  solubility  of  crystals  has  been  shown  to 
be  different  in  different  directions.  Thus,  if  the  surface  of  a  crys- 
talline substance  be  highly  polished  and  then  treated  for  a  short 
time  with  a  suitable  solvent,  faint  patterns,  known  as  etch  figures, 
will  appear  as  a  result  of  the  inequality  of  the  rate  of  solution  in 
different  directions.  When  these  figures  are  examined  under  the 
microscope  the  crystal  form  can  generally  be  determined.  The 
examination  of  etch  figures  has  come  to  be  of  prime  importance  to 
the  metallographer.  Thus,  when  an  appropriate  solvent  is 
applied  to  the  polished  surface  of  an  alloy,  not  only  is  the  crystal 


138  THEORETICAL  CHEMISTRY 

form  revealed  by  the  etch  figures,  but  also  the  presence  of  various 
chemical  compounds  may  be  recognized.  By  a  careful  study 
of  the  etch  figures  developed  on  the  surface  of  highly  polished 
steel,  the  metallographer  may  gather  important  information  as  to 
its  previous  history,  especially  its  heat  treatment. 

Crystal  Form  and  Chemical  Composition.  From  the  pre- 
ceding paragraphs  it  might  be  inferred  that  the  same  substance 
always  assumes  the  same  crystal  form.  While  this  is  true  in  gen- 
eral, there  are  some  substances  which  appear  in  several  different 
crystal  forms.  This  phenomenon  is  termed  polymorphism. 

Calcium  carbonate  is  an  example  of  a  substance  crystallizing  in 
more  than  one  form.  As  calcite,  it  crystallizes  in  the  hexagonal 
system,  while  as  aragonite,  it  crystallizes  in  the  rhombic  system. 
Such  a  substance  is  said  to  be  dimorphous.  Of  the  several  factors 
controlling  polymorphism,  temperature  is  the  most  important. 
Thus  sulphur  crystallizes  at  temperatures  above  95°. 6  in  the  mono- 
clinic  system,  while  at  lower  temperatures  it  assumes  the 
rhombic  form.  The  temperature  at  which  it  changes  from  one 
form  into  the  other  is  termed  its  transition  temperature.  As  has 
been  mentioned  in  an  'earlier  chapter  (p.  14),  some  substances 
may  crystallize  in  the  same  form,  the  characteristic  interfacial 
angles  being  nearly  identical.  Such  substances  are  said  to  be  iso- 
morphous.  This  phenomenon,  discovered  by  Mitscherlich,  has 
been  of  great  use  in  connection  with  the  earlier  investigations  on 
atomic  weights,  as  has  already  been  pointed  out. 

There  can  be  little  doubt  as  to  the  existence  of  an  intimate 
connection  between  crystalline  form  and  chemical  composition. 
Ever  since  the  early  part  of  the  nineteenth  century,  when  Haiiy 
established  the  science  of  crystallography,  various  attempts  have 
been  made  by  chemists  and  crystallographers  to  connect  crystal- 
line form  with  chemical  constitution.  In  1906,  Barlow  and  Pope  * 
made  a  most  notable  contribution  to  the  theories  concerning  the 
relation  between  crystalline  form  and  chemical  constitution. 
Their  ideas  may  be  summarized  as  follows :  —  If  each  atom  be 
considered  as  appropriating  a  certain  space,  called  its  sphere  of 
atomic  influence,  then  (1)  the  spheres  of  atomic  influence  are  so 

*  Jour.  Chem.  Soc.,  91,  1150  (1907). 


SOLIDS  139 

arranged  as  to  occupy  the  smallest  possible  volume  in  every  crystal; 
(2)  the  volumes  of  the  spheres  of  atomic  influence  in  any  substance 
are  proportional  to  the  valences  of  the  constituent  atoms;  (3)  the 
volumes  of  the  spheres  of  influence  of  the  atoms  of  different  elements 
of  the  same  valence  are  nearly  equal,  any  variation  being  in  harmony 
with  their  relations  in  the  Periodic  System.  Barlow  and  Pope 
have  shown  that  the  general  agreement  between  theory  and 
observation  is  most  satisfactory,  a  particularly  strong  argument 
in  favor  of  this  theory  being  the  very  plausible  explanation  which 
it  furnishes  for  a  large  number  of  crystallographic  facts.  It  is 
without  doubt  the  best  working  hypothesis  which  has  yet  been 
offered  for  the  investigation  of  the  dependence  of  crystalline  form 
upon  a  definite  chemical  constitution.* 

Compressibilities  of  the  Solid  Elements.  A  series  of  careful 
measurements  of  the  compressibilities  of  the  elements  by  T.  W. 
Richards  and  his  collaborators,!  has  revealed  the  fact  that  com- 
pressibility is  a  periodic  function  of  atomic  weight.  Richards 
has  advanced  some  interesting  suggestions  as  to  the  importance 
of  compressibility  in  connection  with  intermolecular  cohesion 
and  atomic  volume.  That  there  is  some  relation  between  the 
compressibility,  the  atomic  volumes  and  the  atomic  refractions 
of  the  solid  elements  will  be  apparent  from  the  accompanying 
table.  The  compressibilities  are  expressed  in  megabars  X  106, 
i.e.,  megadynes  per  square  centimeter  X  106. 

Specific  Heats  of  the  Solid  Elements.  As  has  already  been 
stated  in  Chapter  I  (p.  11),  Dulong  and  Petit,  in  1819,  discovered 
the  interesting  fact  that  the  atomic  heats  of  the  solid  elements 
have  a  constant  value  of  6.5.  The  importance  of  this  general- 
ization in  connection  with  the  verification  of  atomic  weights  has 
already  been  pointed  out.  Quite  recently,  Lewis  {  has  directed 
attention  to  the  fact  that  it  is  much  more  rational  to  calculate  the 
atomic  heat  of  an  element  from  the  specific  heat  at  constant 

*  For  a  brief  resume  of  the  somewhat  abstruse  papers  of  Barlow  and  Pope 
the  student  is  referred  to  aAn  Introduction  to  Chemical  Crystallography," 
by  P.  Groth,  translated  by  H.  Marshall. 

t  Zeit.  phys.  Chem.,  61,  77,  100,  171,  183  (1908). 

|  Jour.  Am.  Chem.  Soc.,  29,  1165  (1907). 


140 


THEORETICAL  CHEMISTRY 


COMPRESSIBILITIES,  ATOMIC  VOLUMES  AND  ATOMIC 
REFRACTIONS  OF  THE  ELEMENTS. 


Element. 


Compresa. 


Atomic  Vol. 


Atomic  Refrac- 
tion. 


Lithium 8.8 

Carbon  (diamond) 0.5 

Sodium 15.4 

Magnesium 2.7 

Aluminium 1.3 

Silicon 0.16 

Phosphorus  (red) 9.0 

Phosphorus  (white) 20.3 

Sulphur 12.5 

Potassium 31.5 

Calcium 5.5 

Chromium 0.7 

Manganese 0 . 67 

Iron 0.40 

Nickel 0.27 

Copper 0. 54 

Zinc 1.5 

Cadmium 1.9 

Silver 0.84 

Gold 0.47 

Lead..  2.2 


13.1 

3.4 

23.7 

13.3 


10.1 

11.4 

14.4 

16.6 

15.5 

45.5 

25.3 

7.7 

7.7 

7.1 

6.7 

7.1 

9.5 

13.0 

10.3 

10.2 

18.2 


3.5 

5.0 

4.4 

6.7 

7.7 

7.4 

18.3 

18.3 

16.0 

7.9 

10.0 

15.3 

11.5 

11.6 

9.9 

11.5 

9.8 

13.1 

13.2 

23.1 

24.3 


volume  rather  than  from  the  specific  heat  at  constant  pressure. 
While  it  is  impossible  to  measure  the  specific  heat  at  constant 
volume,  its  value  may  be  derived  from  the  specific  heat  at  constant 
pressure  by  an  application  of  the  laws  of  thermodynamics.  Thus, 
Lewis  has  obtained  the  formula 

TpV 


41.78*' 

where  T  denotes  the  absolute  temperature,  ft  the  coefficient  of 
expansion,  a  the  coefficient  of  compressibility,  Cp  and  Cv  the 
atomic  specific  heats  at  constant  pressure  and  constant  volume 
respectively  and  V  the  atomic  volume.  By  means  of  this  equa- 
tion Lewis  has  established  the  following  generalization:  —  Within 
the  limits  of  experimental  error,  the  atomic  heat  at  constant  volume, 
at  20°  C.,  is  the  same  for  all  solid  elements  whose  atomic  weights  are 
greater  than  that  of  potassium,  and  is  equal  to  5.9.  Whether  this 


SOLIDS 


141 


law  will  be  equally  valid  for  other  temperatures  cannot  be  ascer- 
tained until  the  change  of  compressibility  with  temperature  has 
been  investigated.  The  specific  heat  at  constant  pressure  in  many 
cases  increases  rapidly  with  the  temperature,  but  it  is  also  prob- 
able that  the  term,  Cp  —  Cv,  increases  also.  It  is  therefore  quite 
probable  that  the  specific  heat  at  constant  volume  may  be  inde- 
pendent of  the  temperature. 

Molecular  Volumes  of  Solids.  The  molecular  volumes  of 
solids  are  not  equal  to  the  sum  of  the  atomic  volumes  of  the 
constituents.  Thus  the  molecular  volume  of  potassium  chloride 
is  37.4,  while  the  atomic  volumes  of  potassium  and  chlorine  are 
45.2  and  25.7  respectively,  the  calculated  molecular  volume  being 
70.9.  In  general  the  observed  molecular  volume  is  less  than  the 
calculated  volume.  The  atomic  volume  of  an  element  is  found 
to  be  dependent  upon  the  nature  of  the  compound  from  which  its 
value  is  derived,  two  compounds  of  similar  constitution  giving 
almost  identical  values,  whereas  two  compounds  of  different 
constitution  give  values  which  are  quite  divergent. 

Notwithstanding  the  exhaustive  investigations  of  Kopp  and 
others  in  this  field,  little  progress  has  been  made  toward  the  dis- 
covery of  any  generalizations.  If  the  molecular  volumes  of 
similar  compounds  are  compared,  it  will  be  observed  that  the 
replacement  of  one  element  or  group  by  another  element  or  group, 
produces  a  constant  difference  in  molecular  volume.  This  is 
shown  in  the  following  table  of  molecular  volumes,  where  A  is 
the  difference  between  succeeding  molecular  volumes  in  the 
same  column. 

MOLECULAR  VOLUMES  OF  SOLID  COMPOUNDS. 


A 

A 

KC1.. 

37.4 

NaCl.. 

27.1 

AgCl.. 

25.9 

KBr  

44  3 

6  9 

NaBr 

33  8 

6  7 

AgBr 

31  8 

5  Q 

KI  

54.0 

9.7 

Nal.    . 

43  5 

9  7 

Agl 

42  0 

10  9, 

BaSO4.. 

52.1 

PbSO4  . 

48  0 

SrSO4.  .  . 

46  8 

BaCO3  
Ba(N03)2... 

45.7 
40.8 

6.4 
4.9 

PbCO3  

Pb(N03)2.. 

41.0 
36.8 

7.0 
4.2 

SrCO3... 
Sr(NO,)«... 

40.8 
35.7 

6.0 
5.1 

142  THEORETICAL  CHEMISTRY 

It  will  be  observed  that  the  substitution  of  bromine  for  chlorine, 
or  iodine  for  bromine  in  the  halogen  compounds,  causes  a  nearly 
constant  increase  in  the  molecular  volume,  and  similarly  in  the 
compounds  of  barium,  lead  and  strontium,  when  the  864  group 
is  replaced  by  the  C03  group,  or  when  the  COs  group  is  replaced 
by  the  NO3  group,  an  approximately  constant  difference  in  molec- 
ular volume  results.  Whether  such  constant  differences  may  in 
the  future  afford  a  measure  of  the  relative  affinities  of  the  differ- 
ent atoms  or  groups,  it  is  impossible  to  foretell. 

The  co-volume  of  solids  is  not  constant,  and  consequently  the 
equation 

M/d  =  2  atomic  volumes  +  co-volume, 

which  was  shown  by  Traube  to  apply  to  liquids,  becomes  indeter- 
minate. If  the  co-volume  of  solids  was  constant,  the  above 
equation  would  afford  a  means  of  calculating  molecular  weights 
of  solids.  Up  to  the  present  time  no  satisfactory  method  has  been 
developed  for  the  determination  of  the  molecular  weight  of  a  solid. 
Molecular  Weights  of  Solids.  While  we  do  not  possess  any 
trustworthy  method  for  the  determination  of  molecular  weights 
in  the  solid  state,  an  empirical  formula  derived  by  Longinescu  * 
deserves  mention  as  one  of  the  best  attempts  toward  the  solu- 
tion of  the  problem.  Longinescu  showed  that  the  following 
empirical  relation  is  applicable  to  a  large  number  of  liquids :  — 


in  which  T\  and  772  denote  the  absolute  boiling  points  of  two 
different  liquids,  d\  and  d%  their  densities  at  0°  C.  and  n\  and  n% 
the  numbers  of  atoms  in  their  molecules.  If  the  molecules  of 
two  liquids  contain  the  same  number  of  atoms,  that  is  if  HI  =  n2} 
then  their  absolute  boiling-points  will  be  directly  proportional  to 
their  densities.  Longinescu  tested  this  relation  for  a  series  of 
isomeric  liquids  and  found  it  to  hold  in  many  cases.  Those 
liquids  for  which  the  above  proportionality  does  not  hold  were 
observed  to  be  those  which  the  experiments  of  Ramsay  and 
Shields  had  shown  to  be  associated.  This  suggested  to  Longinescu 
*  Jour.  Chem.  phys.,  i,  289,  296,  391  (1903). 


SOLIDS 


143 


the  possibility  of  using  the  above  relation  to  determine  the  degree 
of  association.  Writing  the  equation  in  the  form 

/T7  fjl 

the  mean  value  of  k  for  non-associated  liquids  was  found  to  be 
100,  while  for  associated  liquids  much  larger  values  were  obtained. 
If  the  equation  be  solved  for  n,  we  have 


A  few  results  for  non-associated  and  associated  liquids  are  given 
in  the  subjoined  tables,  the  agreement  between  the  degree  of 
association  calculated  by  the  above  formula  and  that  determined 
by  Ramsay  and  Shields  from  measurements  of  surface  tension, 
being  satisfactory. 

NON-ASSOCIATED  LIQUIDS. 


Substance. 

n  (known). 

n  (calculated). 

Ethylene  chloride 

8 

8 

Methyl  acetate  .                        .                      . 

11 

12 

Ethyl  acetate                            

14 

14 

Piperidine  .  .             .       .           

17 

18 

Methyl  benzoate  .                      

18 

18 

Triethylamine  .  .             

22 

23 

Decane  

32 

33 

Tetradecane  

44 

46 

ASSOCIATED  LIQUIDS. 


Substance. 

n  (known). 

n  (calculated). 

Methyl  alcohol 

6 

19 

Acetic  acid  . 

8 

14 

Ethyl  alcohol 

9 

19 

Acetone  ...           .    . 

10 

16 

Propyl  alcohol  

12 

20 

Aniline  

14 

19 

Butylamine  

16 

21 

144 


THEORETICAL  CHEMISTRY 


After  having  shown  that  the  above  relation  is  applicable  to  a 
large  number  of  liquids  Longinescu  applied  the  same  formula  to 
solids,  T,  denoting  the  absolute  melting  point  and,  d,  the  density 
at  0°  C.  The  mean  value  of  k  was  found  to  be  70.  When  the 
equation  is  applied  to  certain  solid  organic  compounds,  the  value 
of  7i,  is  found  to  be  identical  with  that  calculated  for  the  same 
substance  in  the  liquid  state.  The  value  of  n  for  inorganic  com- 
pounds is  generally  much  greater  for  substances  in  the  solid  than 
in  the  liquid  state.  A  few  of  the  results  obtained  are  given  in  the 
following  table. 

ASSOCIATION  IN  SOLIDS. 


Substance. 

n  (calculated). 

Water  

29 

Hydrocyanic  acid 

52 

Ammonia 

40 

Cyanogen 

30 

Sulphuric  acid 

9 

Potassium 

60 

Sodium 

56 

Very  little  reliance  is  to  be  placed  upon  the  values  of  n  obtained 
for  solids. 


CHAPTER  VII. 
SOLUTIONS. 

Classification  of  Solutions.  Having  dealt  with  the  properties 
of  pure  substances  in  the  gaseous,  liquid  and  solid  states  we  now 
proceed  to  the  consideration  of  the  properties  of  mixtures  of  two 
or  more  pure  substances.  When  such  a  mixture  is  chemically 
and  physically  homogeneous,  and  no  abrupt  change  in  its  prop- 
erties results  from  an  alteration  of  the  proportions  of  the  com- 
ponents of  the  mixture,  it  is  termed  a  solution.  When  one 
substance  is  dissolved  in  another,  it  is  customary  to  designate  as 
the  solvent  that  component  which  is  present  in  the  larger  proportion, 
the  other  component  being  termed  the  solute.  When  not  more 
than  one-tenth  mol  of  solute  is  present  in  one  liter  of  solution,  the 
solution  is  said  to  be  dilute.  The  detailed  study  of  dilute  solu- 
tions will  be  deferred  until  the  next  chapter. 

There  are  nine  possible  classes  of  solutions,  as  follows:  — 

(1)  Solution  of  gas  in  gas; 

(2)  Solution  of  liquid  in  gas; 

(3)  Solution  of  solid  in  gas; 

(4)  Solution  of  gas  in  liquid; 

(5)  Solution  of  liquid  in  liquid; 

(6)  Solution  of  solid  in  liquid; 

(7)  Solution  of  gas  in  solid; 

(8)  Solution  of  liquid  in  solid; 

(9)  Solution  of  solid  in  solid. 

While  examples  of  all  of  these  different  types  of  solutions  are^known, 
only  the  more  important  classes  will  be  considered  here. 

Solutions  of  Gases  in  Gases.  In  solutions  of  this  class  the 
components  may  be  present  in  any  proportions,  since  gases  are 
completely  miscible.  In  a  mixture  of  gases  where  no  chemical 
action  occurs,  each  gas  behaves  independently,  the  properties  of 

145 


146 


THEORETICAL  CHEMISTRY 


the  gaseous  mixture  being  the  sum  of  the  properties  of  the  con- 
stituents. Thus,  the  total  pressure  of  a  mixture  of  several  gases  is 
equal  to  the  sum  of  the  pressures  which  each  gas  would  exert  were  it 
alone  present  in  the  volume  occupied  by  the  mixture.  This  law  was 
discovered  by  Dalton  *  and  is  known  as  Dalton's  law  of  partial 
pressures.  If  the  partial  pressures  of  the  constituent  gases  be 
denoted  by  pi,  pz,  p3,  etc.,  and  P  and  V  represent  the  total  pres- 
sure and  the  total  volume  of  the  gaseous  mixture,  then 

PV  =  V  (pi  +  p,  +  p3  +  .  .  .  ). 

Dalton's  law  holds  when  the  partial  pressures  are  not  too  great, 

its  order  of  validity  being  the  same 
as  that  of  the  other  gas  laws.  Dal- 
ton's law  can  be  tested  experimen- 
tally by  comparing  the  total  pressure 
of  the  gases  with  the  sum  of  the  pres- 
sures exerted  by  each  gas  before 
mixture.  Van't  Hoff  pointed  out 
the  possibility  of  measuring  the  par- 
tial pressure  of  one  of  the  two  com- 
ponents of  a  gas  mixture,  provided 
a  diaphragm  could  be  found  which 
would  be  pervious  to  one  of  the 
gases  but  not  to  the  other.  It  was 
shown  shortly  afterward  by  Ram- 
say, f  that  the  walls  of  a  vessel  of 
palladium,  when  sufficiently  heated, 
permit  the  free  passage  of  hydrogen 
but  not  of  nitrogen.  The  walls  are 


Hydcogen 


Fig.  48. 


said  to  be  semi-permeable.  A  sketch  of  the  apparatus  used  by 
Ramsay  in  the  verification  of  Dalton's  law  is  shown  in  Fig.  48. 
A  small  vessel  of  palladium,  P,  containing  nitrogen,  is  connected 
with  a  manometer  AB,  which  serves  to  measure  the  pressure  of  the 
gas  in  P.  The  vessel  P  is  enclosed  within  a  larger  vessel  C,  which 
can  be  filled  with  hydrogen  at  known  pressure.  On  heating  P  and 

*  Gilb.  Ann.,  12,  385  (1802). 
t  Phil.  Mag.  (5),  38,  206  (1894). 


SOLUTIONS  147 

passing  a  current  of  hydrogen  at  a  definite  pressure  through  C,  the 
hydrogen  enters  P  until  the  pressures  inside  and  outside  are  equal. 
The  total  pressure  in  P,  measured  on  the  manometer,  is  greater 
than  the  pressure  in  C.  The  difference  between  the  two  pressures 
is  very  nearly  equal  to  the  partial  pressure  of  the  nitrogen.  Con- 
versely, if  a  mixture  of  the  two  gases  be  introduced  into  P,  which  is 
then  heated  and  maintained  at  sufficiently  high  temperature  to  in- 
sure its  permeability  to  hydrogen,  the  partial  pressure  of  the  nitro- 
gen can  be  determined  by  passing  a  current  of  hydrogen  at  known 
pressure  through  C  until  equilibrium  is  attained,  as  shown  by  the 
manometer.  The  difference  between  the  external  and  internal 
pressures  is  the  partial  pressure  of  the  nitrogen.  This  experi- 
ment has  a  very  important  bearing  upon  the  modern  theory  of 
solution. 

Solutions  of  Gases  in  Liquids.  The  solubility  of  gases  in 
liquids  is  limited,  the  extent  to  which  they  dissolve  depending 
upon  the  pressure,  the  temperature,  the  nature  of  the  gas,  and 
the  nature  of  the  solvent.  When  a  liquid  cannot  absorb  any 
more  of  a  gas  at  a  definite  temperature,  it  is  said  to  be  saturated, 
and  the  solution  is  called  a  saturated  solution.  The  solubility  of 
a  gas  in  a  liquid  is  defined  by  Ostwald  as  the  ratio  of  the  volume 
of  the  gas  absorbed  to  the  volume  of  the  absorbing  liquid  at  a 
specified  temperature  and  pressure,  or  if  the  solubility  of  the  gas 
be  represented  by  S,  we  have 

S  =  v/V, 

where  v  is  the  volume  of  gas  absorbed  and  V  is  the  volume  of  the 
absorbing  liquid.  The  " absorption  coefficient"  of  Bunsen  in 
terms  of  which  he  expressed  the  results  of  his  measurements  of 
the  solubility  of  gases,  may  be  defined  as  the  volume  of  a  gas, 
reduced  to  0°  C.  and  76  cm.  pressure  which  is  absorbed  by  unit 
volume  of  a  liquid  at  a  certain  temperature  and  under  a  pressure 
of  76  cm.  of  mercury.  In  certain  cases  the  volume  of  the  gas 
absorbed  is  found  to  be  independent  of  the  pressure,  so  that  if  a 
is  the  coefficient  of  gaseous  expansion,  and  /3  Bunsen's  coefficient 
of  absorption,  then 

S  =  ft  (1+  at). 


148 


THEORETICAL  CHEMISTRY 


The  solubilities  of  a  few  gases  in  water  and  alcohol  as  determined 
by  Bunsen  are  given  in  the  following  table:  — 


Gas. 

Water. 

Alcohol. 

0° 

15° 

0° 

15° 

Hydrogen  
Oxygen 

0.0215 
0.0489 
1.797 

0.0190 
0.0342 
0.1002 

0.0693 
0.2337 
4.330 

0.0673 
0.2232 
3.199 

Carbon  dioxide 

The  solubility  of  gases  in  water  is  appreciably  diminished  by 
the  presence  of  dissolved  solids  or  liquids,  especially  electrolytes. 
Various  theories  have  been  proposed  to  account  for  the  diminished 
solubility  of  gases  in  salt  solutions  but  the  most  satisfactory  is 
that  due  to  Philip,*  who  suggests  that  the  phenomenon  is  caused 
by  the  hydration  of  the  dissolved  salt.  A  portion  of  the  water 
in  the  salt  solution  is  supposed  to  be  in  combination  with  the  salt, 
the  water  which  is  thus  removed  from  the  role  of  solvent,  being 
no  longer  free  to  absorb  gas.  The  solubility  of  a  gas  increases 
with  increase  in  pressure.  For  gases  which  do  not  react  chem- 
ically with  the  solvent,  there  exists  a  simple  relation  between 
pressure  and  solubility,  discovered  by  Henry,  f  This  relation, 
known  as  Henry's  law  may  be  stated  as  follows :  —  When  a  gas  is 
absorbed  in  a  liquid,  the  weight  dissolved  is  proportional  to  the  pressure 
of  the  gas.  Since  pressure  and  volume,  at  constant  temperature, 
are  inversely  proportional  (Boyle's  law),  the  law  of  Henry  may  be 
stated  thus :  —  The  volume  of  a  gas  absorbed  by  a  given  volume  of 
liquid  is  independent  of  the  pressure.  There  is  yet  another  form 
in  which  the  law  may  be  stated  which  is  instructive  in  connection 
with  the  modern  theory  of  solution.  When  a  definite  volume  of 
liquid  is  saturated  with  a  gas  at  constant  temperature  and  pres- 
sure, a  condition  of  equilibrium  is  established  between  the  gas  in 
solution  and  that  in  the  free  space  over  the  solution,  therefore, 
Henry's  law  may  be  stated  as  follows :  —  The  concentration  of  the 
dissolved  gas  is  directly  proportional  to  that  in  the  free  space  above 

*  Trans.  Faraday  Soc.,  3,  140  (1907). 
f  Gilb.  Ann.,  20,  147  (1805). 


SOLUTIONS  149 

the  liquid.  If  Ci  represents  the  concentration  of  the  gas  in  the 
liquid  and  c2  the  concentration  in  the  free  space  above  the  liquid, 
Henry's  law  may  be  expressed  thus:  — 

C1/C2  =  fc, 

where  k  is  known  as  the  solubility  coefficient. 

Dalton  showed  that  the  solubility  of  the  individual  gases  in  a 
mixture  of  gases  is  directly  proportional  to  their  partial  pressures, 
the  solubility  of  each  gas  being  nearly  independent  of  the  presence 
of  the  others. 

As  will  be  seen  from  the  foregoing  table,  the  solubility  of  a  gas 
in  a  liquid  diminishes  with  increase  in  temperature.  Concerning 
the  influence  of  the  nature  of  the  gas  on  its  solubility,  it  may  be 
said  that  those  gases  which  exhibit  acid  or  basic  reactions  are 
the  most  soluble,  the  solubilities  of  neutral  gases  being  small. 
In  the  case  of  many  of  the  very  soluble  gases  Henry's  law  does 
not  hold.  For  example,  ammonia,  a  gas  having  marked  basic 
properties  and  a  large  coefficient  of  solubility,  does  not  obey 
Henry's  law  at  ordinary  temperatures,  the  mass  of  ammonia 
absorbed  not  being  proportional  to  the  pressure.  The  curve 
showing  the  variation  in  solubility  with  pressure  at  0°  C.  has 
two  marked  discontinuities.  At  temperatures  above  100°  C. 
the  gas  obeys  Henry's  law.  Sulphur  dioxide  behaves  similarly, 
the  law  holding  only  for  temperatures  exceeding  40°  C. 

With  regard  to  the  connection  between  the  solvent  power  of  a 
liquid  and  its  nature  but  little  is  known.  About  all  that  can  be 
said  is,  that  the  order  of  solubility  of  gases  in  different  liquids  is 
the  same.  Thus  in  the  preceding  table  the  solubilities  of  hydro- 
gen, oxygen  and  carbon  dioxide  in  water  and  in  alcohol  will  be 
seen  to  be  approximately  proportional.  A  slight  change  in 
volume  always  results  when  a  gas  is  dissolved  in  a  liquid.  In 
general  it  may  be  said  that  the  less  compressible  a  gas  is,  the 
greater  is  the  increase  in  volume  produced  when  it  is  absorbed  by 
a  liquid.  It  is  of  interest  to  note  that  the  increase  in  volume 
caused  by  the  solution  of  a  gas  is  nearly  equal  to  the  value  of  b 
for  the  gas  in  the  equation  of  Van  der  Waals.  This  is  shown  in 
the  following  table :  — 


150  THEORETICAL  CHEMISTRY 


Gas. 

Increase  in  Vol. 

b 

Oxygen                           .  .             .           .... 

0.00115 

0.000890 

Nitrogen      .  .               

0.00145 

0.001359 

Hydrogen           

0.00106 

0.000887 

Carbon  dioxide  

0.00125 

0.000866 

Solutions  of  Liquids  in  Liquids.  Solutions  of  liquids  in  liquids 
can  be  divided  into  three  classes  as  follows: —  (1)  Liquids  which 
are  miscible  in  all  proportions;  (2)  Liquids  which  are  partially 
miscible;  and  (3)  Liquids  which  are  immiscible.  Examples  of 
these  three  classes  in  the  order  mentioned  are,  alcohol  and  water, 
ether  and  water,  and  benzene  and  water.  As  to  the  cause  of 
miscibility  and  non-miscibility  of  liquids  very  little  is  known. 

Partial  Miscibility.  If  a  small  amount  of  ether  is  added  to 
a  large  volume  of  water  in  a  separatory  funnel  and  the  mixture 
vigorously  shaken,  a  perfectly  homogeneous  solution  will  be 
obtained.  On  gradually  increasing  the  amount  of  ether,  shaking 
after  each  addition,  a  concentration  will  eventually  be  reached 
at  which  a  separation  into  two  layers  will  take  place.  The  upper 
layer  is  a  saturated  solution  of  water  in  ether  and  the  lower  layer 
is  a  saturated  solution  of  ether  in  water.  So  long  as  the  relative 
amounts  of  the  two  liquids  is  such  that  the  mixture  does  not 
become  homogeneous  on  standing,  the  composition  of  the  two 
layers  will  be  independent  of  the  relative  amounts  of  the  two 
components.  Measurements  of  the  mutual  solubility  of  liquids 
have  been  made  by  Alexieeff  *  by  placing  weighed  amounts  in 
sealed  tubes  and  observing  the  temperature  at  which  the  mixture 
became  homogeneous.  In  general  the  solubility  of  a  pair  of 
partially  miscible  liquids  increases  with  the  temperature,  and 
therefore  it  may  be  inferred  that  at  a  sufficiently  high  temperature 
the  mixture  will  become  perfectly  homogeneous.  An  example  of 
this  type  of  binary  mixture  is  furnished  by  phenol  and  water,  the 
solubility  curve  of  which  is  shown  in  Fig.  49.  In  this  diagram 
temperature  is  plotted  on  the  axis  of  ordinates  and  percentage 
composition  of  the  solution  on  the  axis  of  abscissae.  Starting 

*  Jour,  prakt.  Chem.,  133,  518  (1882);  Bull.  Soc.  Chem.,  38,  145  (1882). 


SOLUTIONS 


151 


with  a  small  amount  of  phenol  and  adding  it  in  increasing  quan- 
tities to  a  large  volume  of  water,  a  concentration  will  eventually 
be  reached  at  which  the  solution  will  separate  into  two  layers. 
This  concentration  is  represented  by  the  point  A.  On  raising  the 


100  # 


Percentage  Water  in  Phenol 


— \c 


_4 Id 


Percentage  Phenol  in  Water 
Fig.  49. 


loo + 


temperature,  the  solubility  of  phenol  in  water  increases,  as  shown 
by  the  curve  AB.  In  like  manner,  starting  with  pure  phenol 
and  adding  increasing  amounts  of  water,  separation  into  two  layers 
will  occur  at  a  concentration  represented  by  the  point  C.  As  the 
temperature  is  raised  the  solubility  of  water  in  phenol  increases, 
as  shown  by  the  curve  CB.  When  the  temperature  is  raised 
above  68°. 4  C.,  corresponding  to  the  point  B,  phenol  and  water 
become  miscible  in  all  proportions. 

If  we  start  with  a  solution  whose  temperature  and  composition 
is  represented  by  the  point  a,  the  addition  of  increasing  amounts 
of  phenol,  at  constant  temperature  will  be  represented  by  the 
dotted  line  afed.  When  the  point  /  is  reached,  the  solution  will 
separate  into  two  layers  the  composition  of  which  will  be  inde- 
pendent of  the  relative  amounts  of  phenol  and  water.  At  e  the 
solution  will  again  become  homogeneous.  If  the  solution  repre- 


152  THEORETICAL  CHEMISTRY 

sented  by  the  point  a  be  again  chosen  as  the  starting  point,  and  its 
composition  be  kept  unaltered  while  the  temperature  is  raised  to  a 
value  above  68°.4  C.,  the  change  will  be  represented  by  the  dotted 
line  ab.  If  now  the  temperature  be  maintained  constant  and  the 
percentage  of  phenol  increased,  the  alteration  in  composition  will 
be  effected  without  discontinuity,  as  represented  by  the  dotted 
line  be.  On  cooling  the  solution  represented  by  the  point  c  to  the 
initial  temperature  of  a,  the  point  d  will  be  reached.  Thus  it  is 
possible  to  pass  from  a  to  d  by  the  path  abed  without  causing  a 
separation  of  the  components  into  two  layers.  There  is  an 
analogy  between  the  solubility  curve  of  a  pair  of  partially  mis- 
cible  liquids  and  the  dotted,  parabolic  curve  in  the  diagram  of 
the  isothermals  of  carbon  dioxide,  shown  in  Fig.  22.  In  both 
cases  there  is  but  one  phase  outside  of  the  curves,  while  two 
phases  are  coexistent  within  the  area  enclosed  by  the  curves.  The 
analogy  may  be  traced  further,  since  in  each  case  only  one  phase 
can  exist  above  a  certain  temperature.  The  temperature  corre- 
sponding to  the  apex  of  the  parabolic  curve  in  Fig.  22,  is  termed 
the  critical  temperature  of  carbon  dioxide  and  by  analogy  the 
temperature  corresponding  to  the  point,  B,  in  Fig.  49  is  called  the 
critical  solution  temperature.  The  mutual  solubilities  of  some 
pairs  of  partially  miscible  liquids  were  found  by  Alexieeff  to  di- 
minish with  increasing  temperature.  Thus  a  mixture  of  ether  and 
water,  which  is  perfectly  homogeneous  at  ordinary  temperatures, 
becomes  turbid  on  warming.  A  specially  interesting  pair  of 
liquids  is  nicotine  and  water.  At  ordinary  temperatures  these 
liquids  are  miscible  in  all  proportions.  If  the  temperature  is 
raised  above  60°  C.,  the  solution  becomes  turbid  owing  to  incom- 
plete miscibility.  On  continuing  to  heat  the  mixture  the 
mutual  solubility  of  the  liquids  begins  to  increase,  until  at 
210°  C.  they  become  completely  soluble  again.  The  solubility 
relations  of  this  binary  mixture  are  shown  in  Fig.  50.  The  closed 
solubility  curve  defines  the  limits  of  the  coexistence  of  two  layers, 
all  points  outside  of  the  curve  representing  homogeneous  solu- 
tions. 

Complete  Miscibility.     The  study  of  the  vapor  pressures   of 
binary  mixtures  of  completely  miscible  liquids  is  of  great  im- 


SOLUTIONS 


153 


portance  in  connection  with  the  possibility  of  separating  them 
by  the  process  of  distillation.  The  experimental  investigations  of 
Konowalow  *  on  homogeneous  binary  mixtures  of  liquids  have 
shown  that  such  pairs  of  liquids  may  be  divided  into  three  classes 


100  # 


Percentage  Water  in  Nicotine 


210 


Percentage  Nicotine  in  Water 
Fig.  50. 


100  # 


as  follows: — (1)  Mixtures  having  a  maximum  vapor  pressure 
corresponding  to  a  certain  composition,  e.g.,  propyl  alcohol  and 
water;  (2)  Mixtures  having  a  minimum  vapor  pressure  corre- 
sponding to  a  certain  composition,  e.g.,  formic  acid  and  water; 
and  (3)  Mixtures  having  vapor  pressures  intermediate  between 
the  vapor  pressures  of  the  pure  components,  e.g.,  methyl  alcohol 
and  water.  In  considering  the  possibility  of  separating  binary 
mixtures  of  liquids  belonging  to  these  three  classes,  it  is  essential 
to  determine  the  composition  of  both  solution  and  escaping  vapor. 
When  a  pure  liquid  is  boiled  the  composition  of  the  escaping 
vapor  is  the  same  as  that  of  the  liquid  itself,  but  this  is,  in  general, 

*  Wied.  Ann.,  14,  34  (1881). 


154 


THEORETICAL  CHEMISTRY 


not  the  case  when  a  binary  mixture  is  distilled.  The  composition 
of  the  liquid  mixture  in  the  distilling  flask  generally  alters  contin- 
uously when  such  a  mixture  is  distilled. 

(1)  The  relation  between  the  vapor  pressure  and  composition 
of  all  possible  mixtures  of  propyl  alcohol  and  water  is  represented 
graphically  in  Fig.  51.  In  this  diagram  the  compositions  of  the 
mixtures  are  plotted  as  abscissae  and  vapor  pressures  as  ordinates. 
The  vapor  pressures  of  the  pure  components,  water  and  propyl 
alcohol,  at  a  definite  temperature  are  represented  by  A  and  C. 
The  maximum  in  the  vapor-pressure  curve  corresponds  to  a  mix- 


100* 


Water 


Propyl  Alcohol 
Fig.  51. 


100  * 


ture  containing  80  per  cent  of  propyl  alcohol.  The  dotted  curve 
represents  the  boiling-points  of  the  various  mixtures  under  normal 
atmospheric  pressure.  Konowalow  has  shown  that  the  vapor  of 
a  binary  mixture  with  a  minimum  or  maximum  boiling-point  has 
the  same  composition  as  that  of  the  liquid.  The  vapor  of  all 
mixtures  containing  less  than  80  per  cent  of  propyl  alcohol  will 
be  relatively  richer  in  alcohol  than  the  liquid  mixture,  since  the 
vapor  of  propyl  alcohol  is  quite  insoluble  in  water.  If  the  amount 
of  alcohol  in  the  mixture  exceeds  80  per  cent,  then  the  vapor  will 
be  relatively  richer  in  water.  Thus,  whatever  may  be  the  com- 
position of  the  mixture  in  the  distilling  flask,  the  distillate  will 
approximate  to  the  composition  of  the  mixture  having  the  mini- 
mum boiling-point.  The  residue  in  the  flask  will  gradually 


SOLUTIONS 


155 


change  to  pure  water  if  the  original  concentration  were  below 
80  per  cent,  or  to  pure  alcohol  if  the  original  concentration  were 
above  80  per  cent. 

(2)  The  second  type  of  binary  mixture  of  liquids  is  illustrated 
by  formic  acid  and  water,  the  vapor  pressure  and  boiling-point 
curves  for  which  are  shown  in  Fig.  52.  A  mixture  containing 
73  per  cent  of  formic  acid  has  a  minimum  vapor  pressure  and  a 
maximum  boiling-point.  At  this  concentration  the  vapor  and  the 
liquid  have  the  same  composition.  The  vapor  of  mixtures  con- 


100  # 


Water 


formic  Acid- 
Fig.  52. 


100* 


taining  less  than  73  per  cent  of  acid  is  relatively  richer  in  water 
than  the  liquid,  while  the  vapor  of  mixtures  containing  more  than 
73  per  cent  of  acid  contains  relatively  less  water  than  the  liquid. 
Any  mixture  of  formic  acid  and  water  when  distilled  will  thus 
leave  a  residue  in  the  distilling  flask  containing  73  per  cent  of 
acid;  this  residue  will  distil  at  constant  temperature  like  a  homo- 
geneous liquid.  It  was  thought  for  a  long  time  that  such  constant 
boiling  mixtures  were  definite  chemical  compounds  of  the  two 
liquids.  Thus  a  mixture  of  hydrochloric  acid  and  water  contain- 
ing 20.2  per  cent  of  acid  boils  at  110°  C.  under  atmospheric  pres- 
sure. The  composition  of  such  a  mixture  corresponds  very  nearly 
to  the  formula,  HC1.8  H20.  Roscoe  *  showed  that  these  mix- 

*  Lieb.  Ann.  116,  203  (1860). 


156 


THEORETICAL  CHEMISTRY 


tures  are  not  definite  chemical  compounds  since  the  composition 
of  the  distillate  changes  when  the  distillation  is  carried  out  under 
different  pressures. 

(3)  The  vapor-pressure  and  boiling-point  curves  for  methyl 
alcohol  and  water,  a  mixture  typical  of  the  third  class  of  com- 
pletely miscible  liquids,  are  shown  in  Fig.  53,  the  heavy  line 
representing  vapor  pressures  at  65°.2  C.  and  the  dotted  line  the 
boiling  points  under  normal  atmospheric  pressure.  In  this  case 


1005 


Water 


Methyl  Alcohol 
Fig.  53. 


100* 


the  composition  of  both  vapor  and  liquid  alter  continuously  on 
distillation.  The  distillate  will  contain  a  relatively  larger  amount 
of  alcohol  and  the  residue  in  the  distilling  flask,  an  excess  of  water. 
If  this  distillate  be  redistilled  from  a  clean  flask,  a  second  dis- 
tillate still  richer  in  alcohol  will  be  obtained.  By  repeating  this 
process  a  sufficient  number  of  times,  a  more  or  less  complete 
separation  of  the  two  components  of  the  mixture  can  be  effected. 
This  process  is  termed  fractional  distillation. 

Immisdbility.  When  two  immiscible  liquids  are  brought 
together,  the  total  vapor  pressure  is  equal  to  the  sum  of  the  vapor 
pressures  of  the  components;  hence  when  such  a  mixture  is  dis- 
tilled, the  two  liquids  will  pass  over  in  the  ratio  of  their  respective 


SOLUTIONS  157 

vapor  pressures,  the  boiling-point  of  the  mixture  being  the  temper- 
ature at  which  the  sum  of  the  vapor  pressures  of  the  two  liquids 
is  equal  to  the  pressure  of  the  atmosphere.  The  relation  between 
vapor  pressure  and  composition  in  this  case  will  be  represented 
by  a  horizontal  line  drawn  at  a  distance  above  the  axis  of  abscissae 
equal  to  the  sum  of  the  vapor  pressures  of  the  components. 

Nitrobenzene  and  water  may  be  chosen  as  an  example  of  a  pair 
of  liquids  which  are  practically  immiscible.  Under  a  pressure 
of  760  mm.  the  mixture  boils  at  99°  C.  The  vapor  pressure  of 
water  at  this  temperature  is  733  mm.  ;  the  vapor  pressure  of  nitro- 
benzene must  be  760  —  733  =  27  mm.  Notwithstanding  the 
relatively  small  vapor  pressure  of  nitrobenzene  in  the  mixture, 
considerable  quantities  of  it  distil  over  with  the  water.  It  is  this 
fact  that  makes  possible  separations  of  liquids  by  the  process  of 
steam  distillation  so  frequently  employed  by  the  organic  chem- 
ist. The  relative  weights  of  water  and  nitrobenzene  passing 
over  in  a  steam  distillation  may  be  calculated  as  follows  :  —  The 
relative  volumes  of  steam  and  vapor  of  nitrobenzene  which  distil 
over  will  be  in  the  ratio  of  their  respective  vapor  pressures  at  the 
temperature  of  the  experiment,  and  consequently  the  relative 
weights  of  the  two  liquids  which  pass  over  will  be  in  the  ratio, 
Pidi  :  pzdz,  where  pi  and  p2  denote  the  respective  vapor  pressures 
of  water  and  nitrobenzene,  and  di  and  d2  the  corresponding  vapor 
densities.  If  Wi  and  w2  denote  the  weights  of  the  two  liquids  in 
the  state  of  vapor,  then 


or,  since  vapor  density  is  proportional  to  molecular  weight,  we 
may  write 

w  ii  w2  ::  piMi  :  p2M2. 

Substituting  in  this  proportion  the  values  given  above  for  the  vapor 
pressures  of  steam  and  nitrobenzene,  we  have 

wi  :w2  ::733  X  18  :  27  X  123 
or, 

wi  :w2  ::  13,194  :3321. 

Thus  the  weights  of  water  and  nitrobenzene  in  the  distillate  are 
approximately  in  the  ratio  of  4  to  1  notwithstanding  the  fact  that 


158 


THEORETICAL  CHEMISTRY 


the  ratio  of  their  vapor  pressures  at  the  boiling-point  of  the  mix- 
ture is  27  to  1.  If  an  organic  substance  is  not  decomposed  by 
steam,  it  is  possible  to  effect  an  appreciable  purification  by  steam 
distillation,  even  though  its  vapor  pressure  be  relatively  small. 
As  will  be  seen  from  the  above  example,  it  is  the  high  molecular 
weight  of  the  nitrobenzene  which  compensates  for  its  low  vapor 
pressure.  It  is  the  small  molecular  weight  of  water  which  renders 
it  so  suitable  for  steam  distillation. 

Finally,  the  vapor-pressure  and  boiling-point  relations  of  binary 
mixtures  of  partially  miscible  liquids  must  be  considered.     In 


100  # 


Water 


o* 


760  mm.- 


/ 
"C' 


o* 


Isobutyl  Alcohol 
Fig.  54. 


100* 


general  when  two  liquids  are  mixed,  each  lowers  the  vapor  pressure 
of  the  other,  so  that  the  vapor  pressure  of  the  mixture  is  less  than 
the  sum  of  the  vapor  pressures  of  the  components.  As  has  al- 
ready been  pointed  out,  the  composition  of  the  two  layers  in  a 
binary  mixture  of  partially  miscible  liquids  is  independent  of 
the  relative  amounts  of  the  components  present;  hence  the  vapor 
pressure  remains  constant  so  long  as  the  solution  remains  hetero- 
geneous. The  vapor-pressure  and  boiling-point  curves  for  a 
binary  mixture  of  partially  miscible  liquids,  (isobutyl  alcohol  and 
water),  are  shown  in  Fig.  54.  The  horizontal  portion  BC,  repre- 
sents the  vapor  pressures,  at  88°. 5  C.,  of  mixtures  of  isobutyl 
alcohol  and  water  where  two  layers  are  present.  The  vapor 


SOLUTIONS  159 

pressure  of  the  homogeneous  mixtures  are  represented  by  AB  and 
CD,  AB  corresponding  to  solutions  of  isobutyl  alcohol  in  water, 
and  CD  to  solutions  of  water  in  isobutyl  alcohol.  The  dotted 
line  A'E'C'D'  represents  the  boiling-points  of  all  possible  mix- 
tures of  isobutyl  alcohol  and  water,  under  normal  atmospheric 
pressure. 

Solutions  of  Solids  in  Liquids.  The  solubility  of  a  solid  in  a 
liquid  is  limited  and  is  dependent  upon  the  temperature,  the 
nature  of  the  solute  and  the  nature  of  the  solvent.  When  a 
solvent  has  taken  up  as  much  of  a  solute  as  it  is  capable  of  dis- 
solving at  a  definite  temperature,  the  solution  is  said  to  be  satu- 
rated. There  are  two  general  methods  for  the  preparation  of 
saturated  solutions: —  (1)  An  excess  of  the  finely-divided  solute 
is  agitated  with  a  known  amount  of  the  solvent,  at  a  definite 
temperature,  until  equilibrium  is  attained;  (2)  the  solvent  is 
heated  with  an  excess  of  the  solute  to  a  temperature  higher  than 
that  at  which  saturation  is  required,  and  then  cooled  in  contact 
with  the  solid  solute  to  the  desired  temperature.  Both  of  these 
methods  give  equally  satisfactory  results  provided  sufficient 
time  is  allowed  for  the  establishment  of  equilibrium,  and  provided 
the  solid  substance  is  always  present  in  excess.  The  solubility 
of  a  solid  in  a  liquid  may  be  expressed  as  the  number  of  grams  of 
the  solute  in  a  given  mass  or  volume  of  solvent  or  solution,  but 
it  is  usually  expressed  as  the  number  of  grams  of  solute  hi  100 
grams  of  solution.  The  solubility  of  solids  has  recently  been 
shown  to  be  somewhat  dependent  upon  their  state  of  division. 
Thus,  Hulett  *  has  found  that  a  saturated  solution  of  gypsum  at 
25°  C.  contains  2.080  grams  of  CaS04  per  liter,  whereas  when 
very  finely  divided  gypsum  is  shaken  with  this  solution,  it  is 
possible  to  increase  the  content  of  dissolved  CaSOi  to  2.542  grams 
per  liter.  When  a  saturated  solution  is  cooled,  every  trace  of 
solid  solute  being  excluded,  the  excess  of  dissolved  solid  may  not 
separate.  Such  a  solution  is  said  to  be  supersaturated. 

As  a  general  rule  the  solubility  of  solids  in  liquids  increases 
with  the  temperature,  as  shown  in  Fig.  55.  Several  exceptions 
to  this  rule  are  known,  among  which  may  be  mentioned  calcium 
*  Jour.  Am.  Chem.  Soc.,  27,  49  (1905). 


160 


THEORETICAL  CHEMISTRY 


hydroxide,  calcium  sulphate  above  40°  C.,  and  sodium   sulphate 
between  the  temperatures  of  33°  C.  and  100°  C. 

Solubility  curves  are  usually  continuous,  but  exceptions  to  this 
rule  are  common:   the  solubility  curve  of  sodium  sulphate  fur- 


100 


40  60 

>    Temperature 

Fig.  55. 


nishes  an  illustration.  The  discontinuity  in  the  solubility  curve 
of  sodium  sulphate  is  due  to  the  fact  that  we  are  not  dealing  with 
one  solubility  curve,  but  with  two  solubility  curves.  At  temper- 
atures below  33°  C.,  the  dissolved  salt  is  in  equilibrium  with  the 
decahydrate,  Na2S04.10  H20,  whereas  at  temperatures  above 
33°  C.  the  dissolved  salt  is  in  equilibrium  with  the  anhydrous 
salt,  Na2S04.  The  solubility  of  Na2S04.10H2O  increases  with 
the  temperature,  while  the  solubility  of  Na2S04  diminishes.  That 
we  are  actually  dealing  with  two  solubility  curves,  is  proved  by 
the  fact  that  the  solubility  curves  of  the  hydrated  and  anhydrous 
salts  in  supersaturated  solutions  are  continuations  of  the  corre- 
sponding curves  for  saturated  solutions,  as  shown  by  the  dotted 


SOLUTIONS  161 

curves  in  Fig.  55.  If  we  select  any  point,  such  as  p,  lying  between 
a  dotted  curve  and  a  full  curve,  it  is  apparent  that  it  represents  a 
solution  supersaturated  with  respect  to  Na2SO4.10  H20,  but  un- 
saturated  with  respect  to  NasSO^  If  pure  anhydrous  sodium 
sulphate  be  shaken  with  this  solution  it  will  slowly  dissolve,  where- 
as if  a  trace  of  the  hydrated  salt  be  added,  the  solution  will  deposit 
NasSC^.lO  H20,  until  the  amount  remaining  in  solution  corre- 
sponds to  the  solubility  of  the  hydrate  at  that  temperature. 
Supersaturated  solutions  of  some  substances  can  be  preserved 
indefinitely,  provided  all  traces  of  the  solid  phase  are  excluded. 
Such  solutions  are  called  metastable.  On  the  other  hand  there 
are  some  supersaturated  solutions  which  deposit  the  excess  of 
solid  solute  even  when  all  traces  of  it  are  excluded.  These  solu- 
tions are  termed  labile.  The  distinction  between  metastable  and 
labile  solutions  is  not  sharp.  If  a  metastable  solution  is  suffi- 
ciently cooled,  or  if  its  concentration  is  sufficiently  increased,  it 
may  be  made  to  pass  over  into  the  labile  condition.  The  concen- 
tration at  which  this  transition  occurs  is  termed  the  metastable 
limit.  The  stability  of  supersaturated  solutions  has  recently 
been  shown  by  Young  *  to  be  greatly  influenced  by  vibrations  or 
sudden  shocks  within  the  solution.  He  has  been  able  to  control 
the  amount  of  overcooling  in  a  supersaturated  solution,  by  alter- 
ing the  intensity  of  the  vibrations  due  to  the  friction  between  glass 
or  metal  surfaces  within  the  solution. 

Very  little  is  known  concerning  the  relation  between  solubility 
and  the  specific  properties  of  solute  and  solvent. 

Owing  to  the  fact  that  the  change  in  volume  resulting  from  the 
solution  of  a  solid  in  a  liquid  is  very  small,  the  effect  of  pressure 
on  the  solution  is  almost  negligible.  The  chief  factors  condition- 
ing the  change  in  solubility  due  to  increasing  pressure,  are  the 
heat  of  solution  of  the  solute  in  the  nearly  saturated  solution, 
and  the  change  in  volume  on  solidification.  Very  few  experiments 
have  been  made  to  determine  the  effect  of  pressure  on  solubility. 
Van't  Hoff  states  that  the  solubility  of  a  solution  of  ammonium 
chloride,  a  salt  which  expands  when  dissolved,  decreases  by  1  per 
cent  for  160  atmospheres,  while  the  solubility  of  copper  sulphate, 

*  Jour.  Am.  Chem.  Soc.,  33,  148  (1911). 


162 


THEORETICAL  CHEMISTRY 


a  salt  which  contracts  when  dissolved,  increases  by  3.2  per  cent 
for  60  atmospheres. 

Solid  Solutions.  In  general,  when  a  dilute  solution  is  suffi- 
ciently cooled  the  solvent  separates  in  the  form  of  crystals  which 
are  almost  entirely  free  from  the  solute.  When,  however,  the 
temperature  of  a  solution  of  iodine  in  benzene  is  reduced  to  the 
freezing-point,  the  crystals  which  separate  are  found  to  contain 
iodine.  Furthermore,  the  depression  of  the  freezing-point  of  the 
solvent  is  found  to  be  less  than  that  calculated  on  the  assumption 
that  the  solvent  crystallizes  uncontaminated  with  the  solute. 
Such  solutions  were  first  studied  by  Van't  Hoff.*  He  found  that 
when  the  concentration  of  such  abnormal  solutions  is  varied,  the 
ratio  of  the  amount  of  solute  in  the  liquid  solvent  to  the  amount 
of  solute  in  the  solidified  solvent  remains  constant.  Thus,  in  solu- 
tions of  iodine  in  benzene,  the  ratio  of  the  concentration  of  iodine 
in  the  liquid  to  its  concentration  in  the  crystallized  benzene  is 
constant.  In  the  following  table  c\  is  the  concentration  of  iodine 
in  the  liquid  benzene,  and  c2  is  the  concentration  of  iodine  in  the 
solid  benzene. 


C1 

C2 

Cj/Cj 

3.39 
2.587 
0.945 

1.279 
0.925 
0.317 

0.377 
0.358 
0.336 

Van't  Hoff  pointed  out  the  analogy  between  the  distribution  of 
the  solute  between  the  solid  and  liquid  solvent,  and  the  distribution 
of  a  gas  between  a  liquid  and  the  free  space  above  it.  In  other 
words,  the  distribution  follows  Henry's  law  for  the  solution  of  a 
gas  in  a  liquid.  Since  the  crystals  containing  both  solute  and 
solvent  are  perfectly  homogeneous,  Van't  Hoff  suggested  that 
they  be  regarded  as  solid  solutions.  The  mixed  crystals  which 
separate  from  solutions  of  isomorphous  substances  being  chem- 
ically and  physically  homogeneous,  are  to  be  considered  as  solid 
solutions.  Many  alloys  possess  the  properties  characteristic  of 

*  Zeit.  phys.  Chem.,  5,  322  (1890). 


SOLUTIONS  163 

solid  solutions;  hardened  steel,  for  example,  being  regarded  as  a 
homogeneous  solid  solution  of  carbon  in  iron.  One  of  the  char- 
acteristic properties  of  a  dissolved  substance  is  its  tendency  to 
diffuse  into  the  pure  solvent.  Interesting  experiments  performed 
by  Roberts- Austen  *  have  shown  that  even  solids  have  the  prop- 
erty of  mixing  by  diffusion.  Thus,  by  keeping  gold  and  lead  in 
contact  at  constant  temperature  for  four  years,  he  was  able  to 
detect  the  presence  of  gold  in  the  layer  of  lead  at  a  distance  of 
7  mm.  from  the  surface  of  separation.  Many  other  instances  of 
diffusion  in  solids  have  been  observed.! 

Instances  of  gases  and  liquids  dissolving  in  solids  are  also 
known.  Thus  platinum,  palladium,  charcoal  and  other  sub- 
stances have  the  property  of  taking  up  large  volumes  of  hydrogen. 
This  phenomenon,  known  as  occlusion,  is  but  little  understood. 
Van't  Hoff  has  suggested  that  when  hydrogen  dissolves  in  palla- 
dium we  are  really  dealing  with  two  solid  solutions:  one  a  solu- 
tion of  hydrogen  in  palladium  and  the  other  a  solution  of 
palladium  in  solid  hydrogen,  the  system  being  analogous  to 
that  of  two  partially  miscible  liquids. 

Certain  natural  silicates,  the  so-called  zeolites,  are  transparent 
and  homogeneous.  Since  they  contain  varying  quantities  of 
water  they  may  be  regarded  as  examples  of  solutions  of  liquids 
in  solids.  This  classification  is  further  justified  by  the  fact  that 
portions  of  the  water  may  be  removed  and  replaced  by  other 
substances,  such  as  alcohol,  with  apparently  no  change  in  the 
transparency  or  homogeneity  of  the  mineral. 

PROBLEMS. 

1.  2.3  liters  of  hydrogen  under  a  pressure  of  78  cm.  of  mercury,  and 
5.4  liters  of  nitrogen  at  a  pressure  of  46  cm.  were  introduced  into  a  vessel 
containing  3.8  liters  of  carbon  dioxide  under  a  pressure  of  27  cm.    What 
was  the  pressure  of  the  mixture?  Ans.   140  cm.  of  mercury. 

2.  Air  is  composed  of  20.9  volumes  of  oxygen  and  79.1  volumes  of 
nitrogen.    At  15°  C.  water  absorbs  0.0299  volumes  of  oxygen  and  0.0148 

*  Proc.  Roy.  Soc.,  67,  101  (1900). 

t  See  Report  on  Diffusion  in  Solids,  by  C.  H.  Desch,  Chem.  News,  106, 
153  (1912). 


164  THEORETICAL  CHEMISTRY 

volumes  of  nitrogen,  the  pressure  of  each  being  that  of  the  atmosphere. 

Calculate  the  composition  of  the  mixture  of  gases  absorbed  by  the  water. 

Ans.  34.8%  by  vol.  of  O,  and  65.2%  by  vol.  of  N. 

3.  The  vapor  pressure  of  the  immiscible  liquid  system,  aniline-water, 
is  760  mm.  at  98°  C.    The  vapor  pressure  of  water  at  that  temperature  is 
707  mm.    What  fraction  of  the  total  weight  of  the  distillate  is  aniline. 

Ans.   0.28. 

4.  The  boiling-point  of  the  immiscible  liquid  system,  naphthalene-water, 
is  98°  C.  under  a  pressure  of  733  mm.    The  vapor  pressure  of  water  at 
98°  C.  is  707  mm.    Calculate  the  proportion  of  naphthalene  in  the  dis- 
tillate. Ans.  0.207. 


CHAPTER  VIII. 
DILUTE   SOLUTIONS   AND   OSMOTIC   PRESSURE. 

Osmotic  Pressure.  In  the  preceding  chapter  reference  was 
made  to  the  fact  that  diffusion  is  a  characteristic  property  of  solu- 
tions. If  a  few  cubic  centimeters  of  a  concentrated  solution  of 
cane  sugar  are  placed  at  the  bottom  of  a  tall  cylinder,  and  water 
is  added,  care  being  taken  to  prevent  mixture,  the  sugar  immedi- 
ately begins  to  diffuse  into  the  water,  the  process  continuing  until 
the  concentration  of  sugar  is  the  same  throughout  the  liquid. 
The  sugar  molecules  move  from  a  region  of  high  concentration 
to  a  region  of  low  concentration,  the  rate  of  diffusion  being  rela- 
tively slow  owing  to  the  viscosity  of  the  medium.  A  similar 
process  is  encountered  in  the  study  of  gases,  but  the  rate  of  gas- 
eous diffusion  is  extremely  rapid.  In  terms  of  the  kinetic  theory, 
the  movement  of  the  molecules  of  a  gas  from  regions  of  high 
concentration  to  regions  of  low  concentration,  is  to  be  considered 
as  due  to  the  pressure  of  the  gas.  By  analogy,  we  may  regard  the 
process  of  diffusion  in  solutions  as  a  manifestation  of  a  driving 
force,  known  as  the  osmotic  pressure. 

Semi-permeable  Membranes.  The  use  of  a  semi-permeable 
membrane  for  the  measurement  of  the  partial  pressure  of  nitrogen 
in  a  mixture  of  nitrogen  and  hydrogen,  has  already  been  explained. 
A  similar  method  may  be  employed  for  the  measurement  of 
the  osmotic  pressure  of  a  solution,  provided  a  suitable  semi-perme- 
able membrane  can  be  found.  Such  a  membrane  must  prevent 
the  passage  of  the  molecules  of  solute  and  must  be  readily  perme- 
able to  the  molecules  of  solvent;  it  must  exert  a  selective  action 
on  solute  and  solvent.  If  a  solution  is  separated  from  the  pure 
solvent  by  a  semi-permeable  membrane,  diffusion  of  the  solute 
is  no  longer  possible.  Since  equilibrium  of  the  system  can  only 
be  attained  when  the  concentrations  on  both  sides  of  the  mem- 
brane are  equal,  it  follows  that  the  solvent  must  pass  through 

165 


166  THEORETICAL  CHEMISTRY 

the  membrane  and  dilute  the  more  concentrated  solution.  A 
number  of  semi-permeable  membranes  have  been  discovered 
which  are  readily  permeable  to  water  and  nearly,  if  not  entirely, 
impermeable  to  various  solutes.  About  the  middle  of  the  eight- 
eenth century  Abbe  Nollet  discovered  that  certain  animal  mem- 
branes are  permeable  to  water  but  not  to  alcohol. 

Artificial  semi-permeable  membranes  were  first  prepared  by 
M.  Traube.*  If  a  glass  tube,  provided  with  a  rubber  tube  and 
pinch-cock,  be  partially  filled,  by  suction,  with  a  solution  of 
copper  sulphate,  and  then  immersed  in  a  solution  of  potassium 
ferrocyanide,  a  thin  film  of  copper  ferrocyanide  will  be  formed 
at  the  junction  of  the  two  solutions.  When  the  film  has  once 
been  formed,  further  precipitation  of  copper  ferrocyanide  will 
cease,  the  solutions  on  either  side  of  the  film  remaining  clear. 
Traube  showed  that  this  membrane  is  semi-permeable.  He  also 
showed  that  a  number  of  other  gelatinous  precipitates  possess 
the  property  of  semi-permeability.  A  membrane  formed  in  the 
above  manner  is  easily  ruptured  and  is  wholly  inadequate  for 
quantitative  or  even  qualitative  experiments.  Pfeffer  f  devised 
a  method  for  strengthening  the  membrane.  By  depositing  the 
precipitate  in  the  walls  of  a  porous  clay  cup,  the  area  of  unsup- 
ported membrane  is  greatly  diminished  and  its  resisting  power 
correspondingly  increased.  Pfeffer  directs  that  the  cup  to  be 
used  for  this  purpose  must  be  throughly  washed,  and  its  walls 
allowed  to  become  completely  permeated  with  water.  The  cup 
is  then  filled  to  the  top  with  a  solution  of  copper  sulphate,  con- 
taining 2.5  grams  per  liter,  and  allowed  to  stand  for  several  hours 
in  a  solution  of  potassium  ferrocyanide,  containing  2.1  grams 
per  liter.  The  two  solutions  diffuse  through  the  walls  of  the  cup 
and  on  meeting  deposit  a  thin  membrane  of  copper  ferrocyanide. 
When  precipitation  is  complete,  the  cup  is  thoroughly  washed 
and  soaked  in  water.  The  cup  is  then  filled  to  the  top  with  a 
solution  of  cane  sugar,  and  a  rubber  stopper,  fitted  with  a  long 
glass  tube  of  narrow  bore,  is  inserted,  care  being  taken  to  exclude 
air-bubbles.  The  stopper  is  then  made  fast  with  a  suitable 

*  Archiv.  fur  Anat.  und  Physiol,  p.  87  (1867). 
t  Osmotische  Untersuchungen,  Leipzig,  1877. 


DILUTE  SOLUTIONS  AND  OSMOTIC  PRESSURE 


167 


cement,  and  the  cup  completely  immersed  in  a  beaker  of  water. 
The  completed  apparatus  is  shown  in  Fig.  56.  If  the  formation 
of  the  membrane  has  been  successful,  the  level  of  the  liquid  in  the 
vertical  glass  tube  will  slowly  rise  and  will  eventually  attain  a 
height  of  several  meters.  If  the  mem- 
brane is  sufficiently  strong  and  no  leaks 
develop,  the  passage  of  water  through  the 
membrane  will  continue  until  the  hydro- 
static pressure  of  the  column  of  liquid  in 
the  tube  is  great  enough  to  overcome  the 
tendency  of  the  water  to  force  its  way 
into  the  sugar  solution.  As  a  general 
rule,  the  membrane  becomes  ruptured  be- 
fore equilibrium  is  attained. 

Measurement  of  Osmotic  Pressure. 
The  first  direct  measurements  of  osmotic 
pressure  were  made  by  Pfeffer.  His  ex- 
periments deserve  brief  consideration, 
since  the  results  obtained  furnish  th§ 
basis  of  the  modern  theory  of  solution. 
The  cell  used  was  similar  to  that  described 
above,  but  instead  of  employing  a  ver- 
tical glass  tube  as  a  manometer,  the  cup 
was  connected,  as  shown  in  Fig.  57,  with 
a  closed  mercury  manometer.  The  sub- 
stitution of  the  closed  for  the  open  man- 
ometer is  necessitated  by  the  fact,  that 
with  an  open  manometer  so  much  water  entered  the  cell  that 
the  concentration  of  the  solution  became  appreciably  diminished, 
and  the  pressure  actually  measured  corresponded  to  a  solu- 
tion of  smaller  concentration  than  that  introduced  into  the  cell. 
With  the  closed  manometer,  when  a  trace  of  water  has  entered 
the  cell,  sufficient  pressure  is  developed  to  prevent  the  further  en- 
trance of  more  water.  Pfeffer  calculated  that  with  a  cell,  the 
capacity  of  which  was  16  cc.,  the  volume  of  water  entering  before 
equilibrium  was  attained,  did  not  exceed  0.14  cc.  In  his  experi- 
ments, Pfeffer  determined  the  density  of  the  cell  contents  before  and 


Fig.  56. 


168 


THEORETICAL  CHEMISTRY 


after  measurement  of  the  osmotic  pressure,  and  corrected  for  any 
change  in  concentration.  With  this  apparatus  he  made  numer- 
ous measurements  of  the  osmotic 
pressures  of  different  solutions,  the 
entire  apparatus  being  immersed  in 
a  constant-temperature  bath.  With 
solutions  of  cane  sugar  he  obtained 
the  results  given  in  the  accompany- 
ing table,  where  C  denotes  the  per- 
centage concentration  of  the  solution, 
and  P  the  corresponding  osmotic 
pressure,  expressed  in  centimeters  of 
mercury.  The  temperature  varied 
from  13.5°  C.  to  14°.7  C. 


c 

P 

P/C 

1 

53.5 

53.5 

2 

101.6 

50.8 

4 

208.2 

52.0 

6 

307.5 

51.2 

It  is  evident  from  these  results, 
that  the  osmotic  pressure  is  propor- 
\r  tional  to  the  concentration  of  the 
solution,  since  P/C  is  approximately 
constant.    The  deviations  from  con- 
stancy in  the  ratio  of  pressure  to 
\z   concentration  may  be  ascribed  to 
experimental  errors,  since  the  dif- 
ficulties involved  in  these  measure- 
ments are  very  great.     Pfeffer  also 
Fig.  57.  studied  the  influence  of  temperature 

on  osmotic   pressure,  and   showed 

that  as  the  temperature  is  raised  the  pressure  increases.  The 
following  table  gives  his  results  for  a  1  per  cent  solution  of  cane 
sugar. 


DILUTE  SOLUTIONS  AND  OSMOTIC  PRESSURE 


169 


Temperature. 

Osmotic  Pressure. 

6°.  8 

cm. 

50.5 

13°.  2 

52.1 

14°.  2 

53.1 

22°.  0 

54.8 

36°.  0 

56.7 

Osmotic  Pressure  and  the  Nature  of  the  Membrane.  Pfeffer 
also  studied  the  effect  of  the  nature  of  the  membrane  on  osmotic 
pressure.  In  addition  to  copper  ferrocyanide,  he  used  membranes 
of  calcium  phosphate  and  Prussian  blue.  His  results  seemed  to 
indicate  that  the  magnitude  of  the  osmotic  pressure  developed, 
was  dependent  upon  the  nature  of  the  membrane  used. 

The  variations  observed  have  since  been  shown  to  have  been 
due  to  leakage  of  the  calcium  phosphate  and  Prussian  blue  mem- 
branes, the  copper  ferrocyanide  membrane  being  the  only  one 
which  was  capable  of  withstanding  the  pressure.  Ostwald  *  has 
devised  an  ingenious  theoretical  demonstration  of  the  fact  that 
osmotic  pressure  must  be  independent  of  the  nature  of  the  mem- 
brane employed  in  measuring  it.  Let  A  and  B,  in  Fig.  58,  repre- 


Fig.  58. 

sent  two  different  semi-permeable  membranes  placed  in  a  glass 
tube  of  wide  bore.  Let  us  imagine  the  space  between  the  two 
membranes  to  be  filled  with  a  solution,  and  the  tube  immersed 
in  a  vessel  of  water.  If  the  osmotic  pressures  developed  at  A 
and  B  are  pi  and  p%  respectively,  and  pz  is  less  than  p\,  then 
water  will  pass  through  A  until  the  pressure  pi  is  reached.  Since 
the  pressure  at  B  only  reaches  the  value  p2,  however,  the  pres- 
sure pi  can  never  be  attained,  and  a  steady  stream  of  water  from 
A  to  B,  under  the  pressure  pi  —  pz,  will  result.  This,  however, 
would  be  a  perpetual  motion,  and  since  this  is  impossible,  the 
osmotic  pressures  at  the  two  membranes  must  be  the  same". 
*  Lehrb.  d.  allg.  Chem.,  I.,  p.  662. 


170 


THEORETICAL  CHEMISTRY 


Theoretical  Value  of  Osmotic  Pressure.  The  physico-chem- 
ical significance  of  Pfeffer's  results  was  first  perceived  by  Van't 
Hoff.*  In  a  remarkably  brilliant  paper,  he  pointed  out  the 
existence  of  a  striking  parallelism  between  the  properties  of  gases 
and  the  properties  of  dissolved  substances. 

We  have  already  called  attention  to  the  analogy  between  osmotic 
pressure  and  gas  pressure:  we  now  proceed  to  trace  the  connec- 
tion between  osmotic  pressure,  volume  and  temperature,  as  first 
pointed  out  by  Van't  Hoff.  Pfeffer's  experiments  showed  that  at 
constant  temperature,  the  ratio,  P/C,  is  constant  for  any  one 
solute.  Since  the  concentration  varies  inversely  as  the  volume 
in  which  a  definite  amount  of  solute  is  dissolved,  we  obtain,  by 
substituting  l/V  for  C,  the  equation,  PV  =  constant,  which  is 
plainly  the  analogue  of  the  familiar  equation  of  Boyle  for  gases. 
An  examination  of  Pfeffer's  data  for  osmotic  pressures  at  differ- 
ent temperatures,  convinced  Van't  Hoff  that  the  law  of  Gay- 
Lussac  is  also  applicable  to  solutions. 

In  the  following  table,  the  osmotic  pressures  in  atmospheres  for 
a  1  per  cent  solution  of  cane  sugar  at  different  temperatures  are 
recorded,  together  with  the  pressures  calculated  on  the  assump- 
tion that  the  osmotic  pressure  is  directly  proportional  to  the 
absolute  temperature. 


Temperature. 

P  (obs.). 

P  (calc.). 

6°.  8 

0.664 

0.665 

13°.  7 

0.691 

0.681 

15°.  5 

0.684 

0.686 

22°.  0 

0.721 

0.701 

32°.  0 

0.716 

0.725 

36°.  0 

0.746 

0.735 

Since  the  laws  of  Boyle  and  Gay-Lussac  are  both  applicable, 
we  may  write  an  equation  for  dilute  solutions  corresponding  to 
that  already  derived  for  gases,  or 

PV  =  R'T, 

in  which  P  is  the  osmotic  pressure  of  a  solution  containing  a  defi- 
*  Zeit.  phys.  Chem.,  i,  481  (1887). 


DILUTE  SOLUTIONS  AND   OSMOTIC   PRESSURE          171 

nite  weight  of  solute  in  the  volume,  V,  of  solution,  T  being  the 
absolute  temperature  of  the  solution  and  Rf  a  constant  corre- 
sponding to  the  molecular  gas  constant. 

The  molecular  gas  constant  R  has  already  been  evaluated  and 
has  been  found  to  be  equal  to  0.0821  liter-atmosphere. 

Making  use  of  Pfeffer's  data,  Van't  Hoff  calculated  the  value 
of  Rf  in  the  above  equation,  in  the  following  manner:  the  osmotic 
pressure  of  a  1  per  cent  solution  of  cane  sugar  at  0°  C.  is  0.649 
atmosphere,  and  since  the  concentration  of  the  solution  is  1  per 
cent,  the  volume  of  solution  containing  1  mol  of  sugar,  will  be 
34,200  cc.  or  34.2  liters.  Substituting  these  values  in  the  equa- 
tion, we  have 

PV      0.649  X  34.2 
R'  =  _ .  = -— =  0.0813  liter-atmos., 

1  44O 

a  value  which  is  nearly  the  same  as  that  of  the  molecular  gas 
constant,  R.  The  equality  of  R  and  Rr  leads  to  a  conclusion  of 
the  greatest  importance,  as  was  pointed  out  by  Van't  Hoff,  viz., 
"  the  osmotic  pressure  exerted  by  any  substance  in  solution  is  the  same 
as  it  would  exert  if  present  as  a  gas  in  the  same  volume  as  that  occupied 
by  the  solution,  provided  that  the  solution  is  so  dilute  that  the  volume 
occupied  by  the  solute  is  negligible  in  comparison  with  that  occupied 
by  the  solvent. "  It  should  be  remembered  that  we  are  not  justified 
in  concluding  from  this  proposition  of  Van't  Hoff,  that  osmotic 
pressure  and  gaseous  pressure  have  a  common  origin.  While 
the  origin  of  osmotic  pressure  may  be  kinetic,  it  is  also  conceivable 
that  it  may  result  from  the  mutual  attraction  of  solvent  and 
solute,  or  that  it  may  bear  some  relation  to  the  surface  ten- 
sion of  the  solution.  Up  to  the  present  time  no  wholly  satis- 
factory explanation  of  the  cause  of  osmotic  pressure  has  been 
advanced. 

Just  as  1  mol  of  gas  at  0°  C.  and  760  mm.  pressure  occupies  a 
volume  of  22.4  liters,  so  when  1  mol  of  a  substance  is  dissolved 
and  the  solution  diluted  to  22.4  liters  at  0°  C.,  it  will  exert  an 
osmotic  pressure  of  1  atmosphere.  In  other  words,  molar  weights, 
or  quantities  proportional  to  molar  weights,  of  different  substances, 
when  dissolved  in  equal  volumes  of  the  same  solvent  exert  the  same 


172  THEORETICAL  CHEMISTRY 

osmotic  pressure.  If  we  deal  with  n  mols  of  solute  instead  of  1  mol, 
the  general  equation  becomes 

PV  =  nRT. 

But  n  =  g/M,  where  g  is  the  number  of  grams  of  solute  per  liter, 
and  M  is  its  molecular  weight.  Substituting  in  the  preceding 
equation,  we  have 

PV  =  g/M-RT, 
or, 

,.       PV 


Since  P,  V,  g}  R  and  T  are  all  known,  M  can  be  calculated.  The 
direct  measurement  of  the  osmotic  pressure  of  a  solution  does  not 
afford  a  practical  method  for  the  determination  of  the  molecular 
weight  of  dissolved  substances,  because  of  the  experimental 
difficulties  involved  and  the  time  required  for  the  establishment 
of  equilibrium.  There  are  other  and  simpler  methods  for  deter- 
mining molecular  weights  in  solution,  based  upon  certain  proper- 
ties of  solutions  which  are  proportional  to  their  respective  osmotic 
pressures. 

Recent  Work  on  the  Direct  Measurement  of  Osmotic  Pressure. 
It  is  only  within  the  past  decade  that  the  investigations  of  Pfeffer 
have  been  confirmed  and  extended  by  elaborate  and  systematic 
experiments  on  the  direct  measurement  of  osmotic  pressure. 
Morse  and  his  co-workers,*  while  employing  a  method  essentially 
the  same  as  that  of  Pfeffer,  have,  as  the  result  of  much  patient 
labor,  brought  the  apparatus  to  such  a  high  state  of  perfection, 
that  the  experimental  errors  are  now  estimated  to  affect  only 
the  second  place  of  decimals  in  the  numerical  data  expressing 
osmotic  pressures  in  atmospheres.  The  most  important  of  the 
improvements  introduced  by  Morse  are  the  following:  —  (1)  the 
improvement  of  the  quality  of  the  membrane;  (2)  the  improve- 
ment of  the  connection  between  the  cell  and  the  manometer, 
and  (3)  the  improvement  of  the  means  of  accurately  measuring 

*  Am.  Chem.  Jour.,  26,  80  (1901);  34,  1  (1905);  36,  1,  39  (1906);  37,  324, 
425,  558  (1907);  38,  175  (1907);  39,  667  (1908);  40,  1,  194  (1908);  41,  1,  257 
(1909). 


DILUTE  SOLUTIONS  AND   OSMOTIC   PRESSURE 


173 


the  pressure.  The  membrane  of  copper 
ferrocyanide  is  deposited  electrolyti- 
cally.  After  thorough  washing  and 
soaking  in  water,  the  porous  cup,  made 
from  specially  prepared  clay,  is  filled 
with  a  solution  of  potassium  ferrocy- 
anide and  immersed  in  a  solution  of  cop- 
per sulphate.  An  electric  current  is 
then  passed  from  a  copper  electrode 
in  the  solution  of  copper  sulphate,  to 
a  platinum  electrode  immersed  in  the 
solution  of  potassium  ferrocyanide. 
This  drives  the  copper  and  ferrocy- 
anide ions  toward  each  other,  and  the 
membrane  of  copper  ferrocyanide  is 
thus  formed  in  the  walls  of  the  cup. 
The  passage  of  the  current  is  continued 
until  the  electrical  resistance  reaches  a 
value  of  about  100,000  ohms.  The 
cell  is  then  rinsed,  and  soaked  in  water 
for  several  hours,  and  then  the  electro- 
lytic treatment  is  repeated  until  the 
electrical  resistance  attains  a  maximum 
value.  A  solution  of  cane  sugar  is  now 
introduced  into  the  cell,  which  is  con- 
nected with  the  manometer  and  im- 
mersed in  water.  When  the  pressure 
has  attained  its  maximum  value,  the 
apparatus  is  dismantled  and  the  cell, 
after  thorough  washing  and  soaking  in 
water,  is  again  subjected  to  the  electro- 
lytic process  of  membrane  forming.  In 
this  way  the  weak  places  in  the  mem- 
brane which  may  have  yielded  to  the 
high  pressure,  can  be  repaired,  and  by  Fig.  59. 

continued  repetition  of  this  treatment 
the  membrane  can  ultimately  be  brought  to  its  maximum  power 


174 


THEORETICAL  CHEMISTRY 


of  resistance.  A  sketch  of  the  Morse  apparatus  is  shown  in  Fig. 
59.  A  description  of  the  details  of  this  apparatus  lies  beyond  the 
scope  of  this  book.  The  results  of  the  work  of  Morse  and  his 
students  are  of  the  highest  importance.  The  osmotic  pressures 
of  solutions  of  cane  sugar  and  dextrose  have  been  shown  to  be 
proportional  to  the  respective  concentrations,  provided  the  con- 
centration is  referred  to  unit  volume  of  solvent  instead  of  unit 
volume  of  solution.  Thus  in  their  experiments,  the  solutions  were 
made  up  containing  from  0.1  to  1.0  mol  of  solute  in  1000  grams  of 
water.  Morse  calls  such  solutions  weight-normal  solutions  in  con- 
trast to  volume-normal  solutions,  in  which  1  mol  or  a  fraction  of 
a  mol  of  solute  is  dissolved  in  water  and  the  solution  diluted  to  1 
liter.  The  following  data  taken  from  the  work  of  Morse,  shows 
that  when  concentration  is  expressed  on  the  weight-normal  basis, 
there  is  direct  proportionality  between  osmotic  pressure  and  con- 
centration. The  figures  refer  to  solutions  of  dextrose  at  10°  C. 


Molar  Concentration. 

Osmotic  Pressure. 

Per  1000  gm. 
Water. 

Per  Liter  of 
Solution. 

In  Atmos. 

Relative  to  First 
as  Unity. 

0.1 
0.2 
0.5 
1.0 

0.099 
0.196 
0.474 
0.901 

2.39 
4.76 
11.91 
23.80 

1.00 
1.99 
4.98 
9.96 

Morse  and  his  co-workers  also  conclude  from  their  experiments 
at  temperatures  ranging  from  0°  C.  to  25°  C.,  that  the  temper- 
ature coefficients  of  osmotic  pressure  and  gas  pressure  are  prac- 
tically identical.  In  other  words,  their  results  confirm  the 
conclusions  of  Van't  Hoff,  that  the  law  of  Gay-Lussac  is  applicable 
to  solutions.  The  results  of  the  experiments  of  Morse  are  of 
special  interest  in  connection  with  the  proposition  of  Van't  Hoff, 
that  the  osmotic  pressure  of  a  dilute  solution  is  the  same  as  that 
which  the  solute  would  exert  if  it  were  gasified  at  the  same  temper- 
ature and  occupied  the  same  volume  as  the  solution.  The  data 
in  the  following  table  is  taken  from  the  work  of  Morse  on  solutions 


DILUTE  SOLUTIONS  AND   OSMOTIC  PRESSURE 


175 


of  cane  sugar  at  15°  C.  In  addition  to  the  observed  osmotic 
pressures,  the  table  contains  the  corresponding  gas  pressures, 
calculated  (1)  on  the  assumption  that  the  solute  when  gasified 
occupies  the  same  volume  as  the  solution  (proposition  of  Van't 
Hoff),  and  (2)  on  the  assumption  that  it  occupies  the  same  volume 
as  the  solvent  alone. 


Molar  Concentration. 

Osmotic  Pressure  in  Atmos. 

Per  1000  gm.  of 
Water. 

Per  Liter  of 
Solution. 

Obs. 

Calc.  (a). 

Calc.  (b). 

0.1 

0.098 

2.48 

2.30 

2.35 

0.2 

0.192 

4.91 

4.51 

4.70 

0.4 

0.369 

9.78 

8.67 

9.40 

0.6 

0.533 

14.86 

12.51 

14.08 

0.8 

0.684 

20.07 

16.07 

18.79 

1.0 

0.825 

25.40 

19.38 

23.49 

The  calculated  pressures,  recorded  in  the  last  column,  are  in 
much  closer  agreement  with  the  observed  osmotic  pressures,  than 
are  the  calculated  pressures,  recorded  in  the  fourth  column  of  the 
table.  The  proposition  of  Van't  Hoff  should  then  be  modified 
to  read  as  follows :  —  A  dissolved  substance  in  dilute  solution  exerts 
an  osmotic  pressure  equal  to  that  which  it  would  exert  if  it  were  gas- 
.  ified  at  the  same  temperature,  and  the  volume  of  the  gas  were  reduced 
to  that  of  the  solvent  in  the  pure  state.  The  investigations  of  Morse 
and  his  co-workers  may  be  summarized  thus: —  (1)  the  law  of 
Boyle  is  applicable  to  dilute  solutions,  provided  the  concentration 
is  referred  to  1000  grams  of  solvent  and  not  to  1  liter  of  solution; 
(2)  the  law  of  Gay-Lussac  is  also  applicable  to  dilute  solutions, 
that  is  the  temperature  coefficients  of  osmotic  pressure  and  gas 
pressure  are  equal,  and  (3)  the  small  departures  from  the  theo- 
retical values  of  the  osmotic  pressures  may  be  traced  to  hydration 
of  the  solute. 

Direct  measurements  of  the  osmotic  pressure  of  concentrated 
solutions  of  cane  sugar,  dextrose  and  mannite  have  been  made  by 
the  Earl  of  Berkeley  and  E.  G.  J.  Hartley.*  The  method  em- 

*  Proc.  Roy.  Soc.,  73,  436  (1904);  Trans.  Roy.  Soc.  A.,  206,  481  (1906), 


176  THEORETICAL  CHEMISTRY 

ployed  by  these  investigators  is  slightly  different  from  that  of 
Pfeffer  or  Morse;  the  tendency  of  water  to  pass  through  the 
semi-permeable  membrane  is  offset  by  the  application  of  a  counter 
pressure  to  the  solution.  A  membrane  of  copper  ferrocyanide 
is  deposited  electrolytically  very  near  the  outer  surface  of  a  tube 
of  porous  porcelain.  This  tube  is  placed  co-axially  within  a  large 
cylindrical  vessel  of  gun  metal,  an  absolutely  tight  joint  between 
the  two  being  secured  by  an  ingenious  system  of  dermatine  rings 
and  clamps.  The  open  ends  of  the  porcelain  tube  are  closed  by 
rubber  stoppers  fitted  with  capillary  tubes  bent  at  right  angles, 
one  of  the  latter  being  provided  with  a  glass  stop-cock.  When  a 
determination  of  osmotic  pressure  is  to  be  made,  the  apparatus  is 
placed  in  a  horizontal  position  and  water  is  introduced  into  the 
porcelain  tube,  completely  filling  it  and  the  connecting  capillary 
tubes  up  to  a  certain  level.  The  gun  metal  vessel  is  then  filled 
with  the  solution,  and  connected  with  an  auxiliary  apparatus  by 
means  of  which  a  gradually  increasing  hydrostatic  pressure  can 
be  applied.  If  no  pressure  is  applied  to  the  solution,  water  will 
pass  through  the  semi-permeable  membrane  into  the  solution,  and 
the  level  of  the  water  in  the  capillary  tubes  will  fall.  In  carrying 
out  a  measurement,  therefore,  as  soon  as  the  solution  is  introduced 
into  the  gun  metal  vessel,  hydrostatic  pressure  is  applied,  the  mag- 
nitude of  the  pressure  being  so  adjusted  as  to  counterbalance  the 
osmotic  pressure  of  the  solution.  The  level  of  the  water  in  the 
capillary  tubes  serves  to  indicate  the  relative  magnitudes  of 
the  osmotic  and  hydrostatic  pressures.  When  the  level  of  the 
water  in  the  capillary  tubes  remains  constant,  the  two  pressures 
are  in  equilibrium.  The  following  are  the  values  of  the  equilibrium 
pressures  of  solutions  of  cane  sugar,  dextrose  and  mannite  at  0°  C. 
It  must  be  remembered  that  when  the  two  pressures  are  in  equi- 
librium, there  is  always  a  pressure  of  one  atmosphere  on  the  solvent. 
As  will  be  seen  the  pressures  developed  in  the  more  concen- 
trated solutions  are  enormous  and  it  is  a  surprising  fact,  that  even 
in  cases  where  the  highest  pressures  'were  measured,  hardly  a 
trace  of  sugar  was  found  in  the  pure  solvent,  the  membrane 
retaining  its  property  of  semi-permeability  throughout  the  entire 
range  of  pressures.  The  figures  in  the  third  column  are  calculated 


DILUTE  SOLUTIONS  AND  OSMOTIC  PRESSURE 


177 


CANE  SUGAR. 


Osmotic  Pressure  in  Atmospheres. 

Cone.  gm.  per 
Liter. 

Obs. 

Calc. 

180.1 

13.95 

13.95 

300.2 

26.77 

28.74 

420.3 

43.97 

32.55 

540.4 

67.51 

41.85 

660.5 

100.78 

51.16 

750.6 

133.74 

58.14 

DEXTROSE. 


Osmotic  Pressure  in  Atmospheres. 

Cone.  gm.  per 
Liter. 

Obs. 

Calc. 

99.8 

199.5 
319.2 
448.6 
548.6 

13.21 
29.17 
53.19 
87.87 
121.18 

13.21 
26.41 
42.25 
59.28 
72.61 

MANNITE. 


Osmotic  Pressure  in  Atmospheres. 

Cone.  gm.  per 
Liter. 

Obs. 

Calc. 

100 
110 
125 

13.1 
14.6 
16.7 

13.1 
14.4 
16.4 

on  the  assumption  that  there  is  direct  proportionality  between 
osmotic  pressure  and  concentration.  It  is  apparent  that  in  every 
case  the  observed  osmotic  pressure  is  greater  than  the  calculated. 
Even  when  the  concentrations  are  expressed  on  the  weight-normal 


178 


THEORETICAL  CHEMISTRY 


basis,  as  recommended  by  Morse,  the  osmotic  pressure  increases 
more  rapidly  than  the  concentration. 

This  is  well  shown  in  the  accompanying  diagram,  Fig.  60,  due 
to  the  Earl  of  Berkeley.  In  this  diagram,  the  osmotic  pressures 
of  solutions  of  cane  sugar  are  plotted  against  concentrations,  curve  A 
representing  the  actually  observed  osmotic  pressures;  curve  C 


120 


KX) 


180 


i40 


0  100  200  300  400  500  600  700 

Grams  Cane  Sugar  per  liter  of  Solution 

Fig.  60. 

being  traced  on  the  assumption  that  osmotic  pressure  may  be 
calculated  from  the  equation,  PV  =  RT,  where  V  denotes  the 
volume  of  solvent  containing  1  mol  of  cane  sugar;  and  curve  B, 
a  straight  line,  being  drawn  on  the  assumption  that  osmotic 
pressure  may  be  calculated  from  the  equation,  PV  —  RT,  where 
V  represents  the  volume  of  solution  containing  1  mol. 
While  the  theoretical  and  observed  values  of  the  osmotic  pres- 


DILUTE  SOLUTIONS  AND  OSMOTIC   PRESSURE          179 

sure  are  approximately  equal  in  the  more  dilute  solutions,  it  is 
obvious  that  the  observed  values  of  the  osmotic  pressure  of  the 
concentrated  solutions  are  always  greater  than  the  calculated 
values,  even  when  the  calculation  is  made  on  the  assumption  that 
V  in  the  equation,  PV  =  RT,  is  the  volume  of  the  solvent.  The 
abnormally  high  osmotic  pressures  observed  by  the  Earl  of 
Berkeley  have  been  discussed  by  Callendar  *  who  suggests  hydra- 
tion  of  the  solute  as  a  probable  cause. 

He  shows,  that  if  5  molecules  of  water  are  assumed  to  be  asso- 
ciated with  each  molecule  of  cane  sugar  in  the  most  concentrated 
solutions  studied  by  the  Earl  of  Berkeley,  the  discrepancy  between 
the  observed  and  calculated  values  of  the  osmotic  pressure  dis- 
appears. 

Comparison  of  Osmotic  Pressures.  Although  the  difficulties 
involved  in  the  direct  determination  of  osmotic  pressure  are 
many,  these  can  be  avoided  by  the  employment  of  one  of  several 
indirect  methods  which  have  been  devised  for  the  comparison  of 
osmotic  pressures.  All  of  these  methods  depend  upon  the  exchange 
of  water  which  occurs  when  two  solutions  are  separated  by  a 
semi-permeable  membrane.  The  movement  of  the  water  will 
always  be  in  such  a  direction  as  to  tend  to  equalize  the  osmotic 
pressures  on  opposite  sides  of  the  membrane,  or,  in  other  words, 
the  transfer  of  water  will  take  place  from  the  solution  with  the 
lesser  osmotic  pressure  to  the  solution  with  the  greater  osmotic 
pressure. 

The  Plasmolytic  Method.  In  this  method,  solutions  of  various 
substances  are  prepared,  the  concentration  of  each  being  such 
that  its  osmotic  pressure  is  the  same  as  that  of  a  particular  plant 
cell.  Obviously  the  osmotic  pressures  of  all  of  these  solutions 
must  be  equal:  such  solutions  are  said  to  be  isotonic  or  isosmotic. 
The  plasmolytic  method  for  the  comparison  of  osmotic  pressures 
was  developed  by  the  Dutch  botanist,  De  Vries.f  This  method 
depends  upon  the  shrinking  or  swelling  of  the  protoplasmic  sac 
of  plant  cells  when  they  are  immersed  in  a  solution  whose  osmotic 

*  Proc.  Roy.  Soc.  A.,  80,  466  (1908). 

t  Jahrb.  wiss.  Botanik.,  14,  427  (1884);  Zeit.  phys.  Chem.,  2,  415  (1888);  3, 
103  (1889). 


180  THEORETICAL  CHEMISTRY 

pressure  differs  from  that  of  their  own  sap.  De  Vries  found  that 
the  cells  of  Tradescantia  discolor,  Curcuma  rubricaulis,  and  Begonia 
manicata  fulfil  the  necessary  conditions,  viz.;  the  cell  walls  are 
strong  and  resist  alteration  when  immersed  in  solutions,  the  cells 
are  readily  permeable  to  water,  and  the  cell  contents  are  colored, 
thus  enabling  the  slightest  contraction  or  expansion  to  be  de- 
tected. The  cell  walls  are  lined  on  the  inside  with  a  thin,  elastic, 
semi-permeable  membrane  which  encloses  the  colored  contents 
of  the  cell.  The  contents  of  the  cell  consists  of  an  aqueous  solu- 
tion of  several  substances,  among  which  may  be  mentioned 
glucose,  potassium  and  calcium  malate,  together  with  coloring 
matter.  The  osmotic  pressure  of  the  cell  contents  ranges  from 
four  to  six  atmospheres.  The  semi-permeable  membrane  expands 
when  the  contents  of  the  cell  increases  and  contracts  when  the 
contents  diminishes.  In  making  a  comparison  of  osmotic  pres- 
sures by  this  method,  tangential  sections  are  cut  from  the  under 
side  of  the  mid-rib  of  the  leaf  of  one  of  the  above  plants,  e.g., 
Tradescantia  discolor,  and  are  placed  in  the  solution  whose  osmotic 
pressure  it  is  desired  to  compare  with  that  of  the  cell  contents. 
The  cells  are  then  observed  under  the  microscope,  any  decrease 
in  pressure  below  the  normal  resulting  in  a  detachment  of  the 
semi-permeable  membrane  from  one  or  more  points  of  the  cell 
wall.  This  contraction  always  occurs  when  the  cells  are  im- 
mersed in  a  concentrated  solution,  the  phenomenon  being  termed 
plasmolysis.  When  the  solution  in  which  the  cells  are  placed 
has  a  lower  osmotic  pressure  than  the  cell  contents,  no  visible 
effect  is  produced,  the  increased  pressure  within  the  cell  simply 
forcing  the  membrane  closer  to  the  rigid  cell  walls.  By  starting 
with  a  concentrated  solution,  the  osmotic  pressure  of  which  is 
greater  than  that  of  the  cell,  and  gradually  diluting  it,  a  concen- 
tration will  ultimately  be  reached  at  which  the  elastic  membrane 
will  just  completely  fill  the  cell.  This  solution  is  isotonic  with 
the  cell  contents.  In  this  method  the  very  reasonable  assump- 
tion is  made  that  all  of  the  cells  have  the  same  osmotic  pressure, 
any  differences  which  might  have  existed  having  equalized  them- 
selves in  the  living  plant.  The  microscopic  appearance  of  cells 
of  Tradescantia  discolor  when  immersed  in  different  solutions  is 


DILUTE  SOLUTIONS  AND  OSMOTIC  PRESSURE 


181 


shown  in  Fig.  61.  The  appearance  of  the  normal  cell  when  im- 
mersed in  water  or  in  a  solution  whose  osmotic  pressure  is  less 
than  that  of  the  cell  contents,  is  shown  in  A.  When  the  cell  is 
immersed  in  a  0.22  molar  solution  of  cane  sugar  it  appears  as  in 
B,  this  solution  having  a  greater  osmotic  pressure  than  the  cell 
contents.  When  the  cell  is  immersed  in  a  molar  solution  of 


Fig.  61. 

potassium  nitrate,  there  is  marked  plasmolysis,  as  shown  in  C. 
De  Vries  determined  the  concentrations  of  a  large  number  of 
solutions  which  were  isotonic  with  the  cell  contents.  He  ex- 
pressed his  results  in  terms  of  the  isotonic  coefficient,  which  he 
denned  as  the  reciprocal  of  the  molar  concentration.  The  iso- 
tonic coefficient  of  potassium  nitrate  was  taken  equal  to  3.  A 
few  of  De  Vries*  results  are  given  in  the  following  table. 


Substance. 

Formula. 

Isotonic 
Coefficient. 

Glycerol  

^ 

C3H803 

C6H1206 

Ci2H22On 

C4H605 
C4H606 
C6H807 
KN03 
MgCl2 

1.78 
1.81 
1.88 
1.98 
2.02 
2.02 
3.00 
4.33 

Glucose  

Cane  sugar.  .  .  . 

Malic  acid  
Tartaric  acid.  . 

Citric  acid.  .  .  . 

Potassium  nitrj 
Magnesium  chl 

ite  

oride 

182  THEORETICAL  CHEMISTRY 

De  Vries  applied  the  plasmolytic  method  to  the  determination 
of  the  molecular  weight  of  raffinose.  At  that  time  there  was 
considerable  uncertainty  as  to  the  correct  formula  of  crystallized 
raffinose,  three  different  formulas,  all  consistent  with  the  results 
of  analysis,  having  been  proposed  as  follows:  —  Ci8H32Oi6. 
5  H2O,  Ci2H22Oii.3  H2O,  and  CseH^C^.  De  Vries  found  that  a 
3.42  per  cent  solution  of  cane  sugar  was  isotonic  with  a  5.96  per 
cent  solution  of  raffinose.  Letting  the  unknown  molecular 
weight  of  raffinose  be  represented  by  M ,  then 

3.42:5.96  ::342:M. 

Solving  the  proportion  we  have,  M  =  596.  This  result  has  since 
been  confirmed  by  purely  chemical  methods,  and  the  formula, 
Ci8H320i6.5  H20,  the  molecular  weight  of  which  is  594,  is  thus 
established. 

The  Blood  Corpuscle  Method.  The  red  blood  corpuscle  is  a 
cell,  the  contents  of  which  is  enclosed  by  a  thin  elastic  semi- 
permeable  membrane.  Unlike  the  plant  cells,  there  is  no  resistant 
cell  wall  to  give  support  to  the  membrane,  so  that  when  red  blood 
corpuscles  are  immersed  in  water  they  at  first  swell,  owing  to  the 
osmotic  pressure  developed,  and  finally  burst.  When  the  mem- 
brane is  ruptured,  the  coloring  matter  of  the  cell,  the  haemoglobin, 
escapes  and  the  water  acquires  a  deep  red  color. 

Advantage  of  this  behavior  of  red  blood  corpuscles  was  taken 
by  Hamburger  *  for  the  comparison  of  osmotic  pressures.  He 
found  that  when  a  1.04  per  cent  solution  of  potassium  nitrate 
is  added  to  the  defibrinated  blood  of  a  bullock,  the  corpuscles 
will  settle  completely  to  the  bottom,  while  the  supernatant  liquid 
will  remain  clear.  On  the  other  hand,  if  a  0.96  per  cent  solution 
of  potassium  nitrate  is  used,  the  corpuscles  will  not  settle  and  the 
supernatant  liquid  becomes  colored.  If  more  dilute  solutions  of 
potassium  nitrate  are  used,  the  solution  acquires  a  still  deeper 
color.  By  careful  adjustment,  a  concentration  of  potassium 
nitrate  can  be  found  in  which  the  red  blood  corpuscles  will  just 
settle.  In  like  manner,  the  concentration  of  solutions  of  other 
substances  can  be  so  adjusted  as  to  cause  the  precipitation  of  the 

*  Zeit.  phys.  Chem.,  6,  319  (1890). 


DILUTE  SOLUTIONS  AND   OSMOTIC  PRESSURE          183 

corpuscles.  These  solutions  are  isotonic.  Without  going  into 
details,  it  may  be  said  that  the  isotonic  coefficients  obtained  by 
Hamburger,  agree  well  with  those  obtained  by  the  plasmolytic 
method. 

The  Haematocrit  Method.  In  this  method  developed  by 
Hedin,*  advantage  is  again  taken  of  the  properties  of  red  blood 
corpuscles.  As  has  already  been  stated,  when  red  blood  cor- 
puscles are  immersed  in  solutions  of  gradually  diminishing 
concentration  of  the  same  solute,  they  continue  to  swell  and  ulti- 
mately the  semi-permeable  envelope  bursts.  On  the  other  hand, 
when  the  corpuscles  are  immersed  in  solutions  of  gradually  increas- 
ing concentration,  they  shrink,  owing  to  the  transfer  of  water 
from  the  corpuscles.  It  is  apparent  that  there  must  be  a  certain 
concentration  for  each  solute  which  will  cause  no  change  in  the 
volume  of  the  corpuscles.  To  determine  this  concentration,  use 
is  made  of  an  instrument  known  as  an  hcematocrit.  This  is 
simply  a  graduated  thermometer-tube  which  may  be  attached  to 
the  spindle  of  a  centrifugal  machine.  When  the  spindle  is  re- 
volved at  high  speed,  the  corpuscles  collect  in  the  bottom  of  the 
graduated  tube.  A  measured  volume  of  blood  is  centrifuged 
until  no  further  shrinkage  in  volume  of  the  corpuscles  can  be 
detected  in  the  haematocrit.  The  same  volume  of  blood  is  then 
added  to  each  of  a  series  of  solutions  whose  concentration  dimin- 
ishes progressively,  and  the  volume  of  the  corpuscles  is  determined 
as  in  pure  blood.  In  this  way  the  concentration  of  the  solution 
is  found,  in  which  the  volume  of  the  corpuscles  is  the  same  as  in 
the  undiluted  blood.  By  proceeding  in  a  similar  manner  with 
solutions  of  different  substances,  a  series  of  isotonic  coefficients 
can  be  determined.  The  following  table  gives  a  comparison  of 
the  isotonic  coefficients  of  various  substances  obtained  by  the 
plasmolytic,  blood  corpuscle  and  haematocrit  methods.  The  iso- 
tonic coefficients  are  referred  to  that  of  cane  sugar  as  unity. 

There  are  other  methods  which  may  be  used  for  the  comparison 
of  osmotic  pressures,  among  which  may  be  mentioned  that  due  to 
Wladimiroff,f  involving  the  use  of  bacteria,  and  the  interesting 

*  Ibid.,  17,  164  (1895). 

t  Zeit.  phys.  Chem.,  7,  529  (1891). 


184 


THEORETICAL  CHEMISTRY 


Substance. 

Plasmolytic 
Method. 

Corpuscle 
Method. 

Hsematocrit 
Method. 

1  00 

1  00 

1  00 

MgSO4.  . 

1  09 

1  27 

1    10 

KNO3 

1  67 

1  74 

1   84 

NaCL. 

1  69 

1  75 

1  74 

CH3.COOK 

1  67 

•    1  66 

1  67 

CaCl2  

2  40 

2  36 

2  33 

method  developed  by  Tammann,*  in  which  artificially  prepared 
membranes  are  employed. 

Osmotic  Pressure  and  Diffusion.  That  there  is  a  very  close 
connection  between  osmotic  pressure  and  diffusion,  has  already 
been  pointed  out.  In  fact  the  osmotic  pressure  of  a  solution  may 
be  regarded  as  the  driving  force  which  causes  the  molecules  of  a 
dissolved  substance  to  distribute  themselves  uniformly  through- 
out the  solution. 

The  process  of  diffusion  was  first  systematically  investigated 
by  Graham  f  hi  1850,  but  it  was  not  until  five  years  later  that 
the  general  law  of  diffusion  was  enunciated  by  Fick.J  He  proved 
theoretically  and  experimentally  that  the  quantity  of  solute,  ds, 
which  diffuses  through  an  area  A,  in  a  time  dt,  when  the  concen- 
tration changes  by  an  amount  dc,  in  a  distance  dx,  at  right 
angles  to  the  plane  of  A,  is  given  by  the  equation 

ds=  -DA^-dt, 
dx 

in  which  D  is  a  constant,  known  as  the  coefficient  of  diffusion. 
Interpreting  the  equation  of  Fick  in  words,  we  see  that  the  coeffi- 
cient of  diffusion  is  the  amount  of  solute  which  will  cross  1  square 
centimeter  in  1  second,  if  the  change  of  concentration  per  centi- 
meter is  unity. 

The  phenomena  of  diffusion  have  also  been  investigated  by 

*  Wied.  Ann.,  34,  299  (1888). 

f  Phil.  Trans.  (1850),  p.  1,  805;   (1851),  p.  483. 

J  Pogg.  Ann.,  94,  59  (1855). 


DILUTE  SOLUTIONS  AND   OSMOTIC  PRESSURE          185 

Nernst  *  and  Planck,  f  If  we  have  a  tall  cylindrical  vessel  con- 
taining a  solution  of  a  non-electrolyte  in  its  lower  part,  and  pure 
water  at  the  top,  the  solute  will  slowly  diffuse  upward  into  the 
water. 

Assuming  the  osmotic  pressure  at  a  height  x,  to  be  P,  and  letting 
A  denote  the  area  of  cross-section  of  the  cylinder,  the  solute  in 
the  layer  whose  volume  is  A  dx,  will  be  subjected  to  a  force  equal 
to  —  A  dP,  the  negative  sign  indicating  that  the  force  acts  in  the 
direction  of  diminishing  pressure.  If  c  is  the  concentration  in 
mols  per  cubic  centimeter,  the  force  acting  on  each  molecule  in 
this  layer  will  be 

_A^fdP  =  _  l^dP 
cA    dx  c    dx 

Let  F  denote  the  force  necessary  to  drive  a  single  molecule  through 
the  solution  with  the  velocity  of  one  centimeter  per  second.  Since 
the  velocity  is  constant,  the  resistance  due  to  the  viscosity  of  the 
medium  must  also  be  denoted  by  F.  The  velocity  attained  will 
be 

_!  ^ 
cF'  dx' 

If  dN  represents  the  number  of  molecules  crossing  each  layer  in 
a  time  dt,  then,  since  the  number  crossing  unit  area  per  second 
is  proportional  to  the  concentration  and  to  the  mean  velocity  of 
the  molecules,  we  shall  have 

,A,  1    dP  A 

dN  =  --  ^--T-Acdt, 
cF   dx 

or, 

,A,  1  AdP  ,t 

dN  =  —  -^A-^-dt. 
F     dx 

When  the  solution  is  dilute,  we  may  apply  the  general  equation, 
PV  =  RT,  remembering  that  V  =  1/c.  Substituting  in  the  pre- 
ceding equation,  we  have 


-. 

F       dx 

*  Zeit.  phys.  Chem.,  2,  40,  615  (1888);  4,129  (1889). 
t  Wied.  Ann.,  40,  561  (1890). 


186  THEORETICAL  CHEMISTRY 

Comparing  this  equation  with  that  of  Fick,  we  see  that  the 
coefficient  of  diffusion  D,  corresponds  to  the  factor,  RT/F.  From 
the  equation  of  Nernst  it  is  possible  to  calculate  the  force  required 
to  drive  a  molecule  of  solute  through  the  solution  with  unit  veloc- 
ity. Thus,  solving  the  above  equation  for  F,  we  have 

RT.dc, 

~  MAtedt- 

By  means  of  this  equation,  it  has  been  calculated  that  the  force 
necessary  to  drive  one  molecule  of  formic  acid  through  water 
with  a  velocity  of  one  centimeter  per  second  at  0°  C.  is  equal  to 
the  weight  of  4,340,000,000  kilograms.  It  is  difficult  at  first  to 
realize  that  such  enormous  forces  are  operative  in  solutions,  but 
when  one  considers  the  minute  size  of  the  molecules  and  the  great 
resistance  offered  by  the  medium,  it  becomes  evident  that  a  very 
large  driving  force  must  be  applied  to  produce  an  appreciable 
movement  of  the  solute  through  the  solution. 

Principle  of  Soret.  If  a  solution  is  maintained  at  a  uniform 
temperature  it  will  ultimately  become  homogeneous;  if,  on  the 
other  hand,  two  parts  of  a  homogeneous  solution  are  kept  at 
different  temperatures  for  some  time,  the  solution  will  become 
more  concentrated  in  the  colder  portion.  This  phenomenon  was 
first  investigated  by  Soret.*  The  experiments  of  Soret  are  of 
special  interest,  since  they  furnish  a  means  of  determining  the 
influence  of  temperature  on  osmotic  pressure.  Thus,  if  the  law 
of  Gay-Lussac  holds  for  osmotic  pressure,  the  colder  portion  of  a 
solution  should  increase  in  concentration  by  1/273  for  each  degree 
of  difference  in  temperature.  The  experimental  results  are  in 
satisfactory  agreement  with  the  requirements  of  theory,  and  con- 
stitute another  proof  of  the  applicability  of  the  gas  laws  to  dilute 
solutions. 

Lowering  of  Vapor  Pressure.  It  has  long  been  known  that 
the  vapor  pressure  of  a  solution  is  less  than  that  of  the  pure  sol- 
vent, provided  the  solute  is  non-volatile.  The  investigations  of 
von  Babo  and  Wiillner  f  on  the  lowering  of  vapor  pressure  of 

*  Ann.  Chem.  Phys.  (5),  22,  293  (1881). 
f  Fogg.  Ann.,  103,  529  (1858). 


DILUTE  SOLUTIONS  AND  OSMOTIC   PRESSURE 


187 


various  liquids  when  non-volatile  substances  are  dissolved  in 
them,  resulted  in  the  following  generalizations: —  (1)  The  lower- 
ing of  the  vapor  pressure  of  a  solution  is  proportional  to  the  amount 
of  solute  present',  and  (2)  For  the  same  solution  the  lowering  of  the 
vapor  pressure  at  constant  temperature  is  the  same  fraction  of  the 
vapor  pressure  of  the  pure  solvent. 

In  1887,  Raoult,*  as  the  result  of  an  exhaustive  experimental 
investigation,  enunciated  the  following  laws: — (1)  When  equi- 
molecular  quantities  of  different  non-volatile  solutes  are  dissolved  in 
equal  volumes  of  the  same  solvent,  the  vapor  pressure  of  the  solvent  is 
lowered  by  a  constant  amount;  and  (2)  The  ratio  of  the  observed 
lowering  of  the  vapor  pressure  to  the  vapor  pressure  of  the  pure  sol- 
vent is  equal  to  the  ratio  of  the  number  of  mols  of  solute  to  the  total 
number  of  mols  in  the  solution.  The  ratio  of  the  observed  lowering 
to  the  original  vapor  pressure  is  called  the  relative  lowering  of  the 
vapor  pressure.  Letting  pi  amd  p2  denote  the  vapor  pressures  of 
solvent  and  solution,  Raoult 's  second  law  may  be  put  in  the 
form 

Pi-Pz  =      n 
pi          N  +  n' 

in  which  n  and  N  represent  the  number  of  mols  of  solute  and 
solvent  respectively.  Some  of  Raoult's  results  for  ethereal  solu- 
tions are  given  in  the  accompanying  table. 


PI-PJ 

Mols  of  Solute 

Pi  —  Pz 

PI 

Substance. 

per  100  mols 
of  Solution. 

Pi 
for  Solution. 

for  1  molar 
per  cent 
Solution. 

Turpentine  

8.95 

0.0885 

0.0099 

Methyl  salicylic  acid  

2.91 

0.026 

0.0089 

Methyl  benzole  acid  

9.60 

0.091 

0.0095 

Benzole  acid  

7.175 

0.070 

0.0097 

Trichloracetic  acid 

11  41 

0  120 

0  0105 

Aniline                  

7  66 

0  081 

0  0106 

The  results  given  in  the  fourth  column  of  the  table  are  nearly 

*  Compt.  rend.,  104,  1430  (1887);  Zeit.  phys.  Chem.,  2,  372  (1888);  Ann. 
Chem.  Phys.  (6),  15,  375  (1888). 


188  THEORETICAL  CHEMISTRY 

constant,  and  are  in  close  agreement  with  the  theoretical  value 
of  the  relative  lowering  of  a  1  molar  per  cent  solution  calculated 
as  follows  :  — 


i  f\r\ 

Pi          100  +  1 

When  the  solution  is  very  dilute,  the  number  of  mols  of  solute 
is  negligible  in  comparison  with  the  number  of  mols  of  solvent, 
and  the  equation  of  Raoult  may  be  written 

Pi  -  Pz       n_ 

Pi  AT* 

Since  n  =  g/m,  and  N  =  W/M,  where  g  and  W  are  the  weights  of 
solute  and  solvent  respectively,  and  ra  and  M  are  the  correspond- 
ing molecular  weights,  the  above  equation  becomes 

Pi-  PZ  =  gM 
Pi          Wm 

This  equation  enables  us  to  calculate  the  molecular  weight  of  a 
dissolved  substance  from  the  relative  lowering  of  the  vapor 
pressure  produced  by  the  solution  of  a  known  weight  of  solute  in 
a  known  weight  of  solvent.  Solving  the  equation  for  m,  we 
have 


W      pi-  pz 

As  an  illustration  of  the  application  of  this  equation,  we  may 
take  the  determination  of  the  molecular  weight  of  ethyl  benzoate 
from  the  following  experimental  data:  —  The  vapor  pressure  at 
80°  C.  of  a  solution  of  2.47  grams  of  ethyl  benzoate  in  100  grams 
of  benzene  is  742.6  mm.:  the  vapor  pressure  of  pure  benzene  at 
80°  C.  is  751.86  mm.  Substituting  in  the  equation,  we  have 

_  2.47  X  78  751.86 

100       '  751.86  -  742.6 

The  molecular  weight  calculated  from  the  formula,  C6H5COO.C2H6, 
is  150. 

The  difficulties  which  attend  the  accurate  measurement  of  the 
vapor  pressure  of  a  solution  by  the  static  method  have  already 


DILUTE  SOLUTIONS  AND  OSMOTIC  PRESSURE 


189 


been  mentioned.  While  there  are  other  methods  which  are  pref- 
erable for  the  determination  of  the  molecular  weight  of  dissolved 
substances,  the  vapor  pressure  method  has  one  marked  advan- 
tage, —  it  can  be  used  for  the  same  solution  at  widely  divergent 
temperatures.  The  method  devised  by  Walker  and  already 
described  in  connection  with  the  determination  of  the  vapor 
pressure  of  pure  liquids  (p.  92)  is  well  adapted  to  the  measure- 
ment of  the  vapor  pressure  of  solutions. 

Connection  between  Lowering  of  Vapor  Pressure  and  Osmotic 
Pressure.  The  relation  between  osmotic  pressure  and  the 
lowering  of  vapor  pressure  has  been  derived 
in  the  following  manner  by  Arrhenius.* 
Imagine  a  very  dilute  solution  contained  in 
the  wide  glass  tube  A,  Fig.  62.  The  tube, 
A,  is  closed  at  its  lower  end  with  a  semi- 
permeable  membrane,  and  dips  into  a  vessel, 
5,  which  contains  the  pure  solvent.  The 
entire  apparatus  is  covered  by  a  bell-jar  C, 
and  the  enclosed  space  exhausted.  Let 
h  be  the  difference  in  level  between  the 
solvent  and  solution  when  equilibrium  is 
established,  that  is,  when  the  hydrostatic 
pressure  of  the  column  of  liquid  is  equal  to 
the  osmotic  pressure.  When  equilibrium  is  . 
attained,  the  vapor  pressure  of  the  solution 
at  h  will  be  equal  to  the  vapor  pressure  of 
the  solvent  at  this  point.  If  the  vapor  pressure  of  the  pure  sol- 
vent is  pi,  and  p2  Is  the  vapor  pressure  of  the  solution  at  h,  we 
shall  have 

Pi  -  Pz  =  M,  (1) 

where  d  denotes  the  density  of  the  vapor.  Let  v  be  the  volume  of 
1  mol  of  solvent  in  the  state  of  vapor,  then 

Pit;  =  RT, 
and 

RT 

v  = 

Pi 

*  Zeit.  phys.  Chem.,  3,  115  (1889). 


B 


Fig.  62. 


190  THEORETICAL  CHEMISTRY 

If  the  molecular  weight  of  the  solvent  is  M,  we  may  replace  v 
by  M/d,  when  the  preceding  equation  becomes 

M^^RT 
d  "  pi  ' 
or, 


The  solution  being  very  dilute  the  osmotic  pressure  may  be  cal- 
culated from  the  equation 

PV  =  nRT, 

where  P  is  the  osmotic  pressure  of  the  solution,  V  the  volume 
containing  1  mol  of  solute,  and  n  the  number  of  mols  of  solute 
present.  If  s  represents  the  density  of  the  solvent  and  also  of 
the  solution,  since  it  is  very  dilute,  we  may  write 

P  =  hs, 
and 


where  g  is  the  number  of  grams  of  the  solvent  in  which  the  n 
mols  of  solute  are  dissolved.  Substituting  these  values  of  P  and 
V  in  the  general  equation,  we  have 

PV  =  nRT  =  hg, 
and  solving  for  h, 

(3) 


Substituting  the  values  of  d  and  h,  given  in  equations  (2)  and 
(3),  in  equation  (1)  we  have 


nRT 

»-*-—  ••--  r-         (4) 

Rearranging  equation  (4),  and  remembering  that  N  =  g/M,  we 
obtain 

Pi-  P*      ^  ff{\ 

~^T   =  r 

This  equation  it  will  be  seen,  is  identical  with  that  derived  experi- 
mentally by  Raoult  for  very  dilute  solutions. 


DILUTE  SOLUTIONS  AND   OSMOTIC   PRESSURE          191 

Van't  Hoff  showed,  by  an  application  of*  thermodynamics  to 
dilute  solutions,  that  the  relation  between  osmotic  pressure  and 
the  relative  lowering  of  the  vapor  pressure  is  expressed  by  the 
equation 

Pi  -  pz      MP 
pi          sRT' 

in  which  the  symbols  have  the  same  significance  as  above.     This 
equation  may  be  reconciled  easily  with  the  equation  of  Raoult. 
If  n  in  equation  (5)  be  replaced   by  its  equal,  PV/RT,  the 
equation  becomes 

Pi  -  pa 


= 

PI       NET' 

But  7  =  NM/s,  hence 

pi-7*»  =  MP 
pi          sRT' 

This  equation  shows  that  the  relative  lowering  of  the  vapor  pressure 
is  directly  proportional  to  the  osmotic  pressure. 

Elevation  of  the  Boiling-Point.  Just  as  the  vapor  pressure  of 
a  solution  is  less  than  that  of  the  pure  solvent,  so  the  boiling-point 
of  a  solution  is  correspondingly  higher  than  the  boiling-point  of 
the  solvent.  It  follows  that  when  equimolecular  quantities  of 
different  substances  are  dissolved  in  equal  volumes  of  the  same 
solvent,  the  elevation  of  the  boiling-point  is  constant.  Thus,  the 
molecular  weight  of  any  soluble  substance  may  be  determined  by 
comparing  its  effect  on  the  boiling-point  of  a  particular  solvent, 
with  that  of  a  solute  of  known  molecular  weight.  The  elevation 
in  boiling-point  produced  by  dissolving  1  mol  of  a  solute  in  100 
grams,  or  100  cubic  centimeters,  of  a  solvent  is  termed  the  molec- 
ular elevation,  or  boiling-point  constant  of  the  solvent.  In  deter- 
mining the  boiling-point  constant  of  a  solvent,  a  fairly  dilute 
solution  is  employed  and  the  elevation  in  the  boiling-point  is 
observed;  the  value  of  the  constant  is  then  calculated  on  the 
assumption  that  the  elevation  in  boiling-point  is  proportional  to 
the  concentration. 

If  g  grams  of  a  substance  of  unknown  molecular  weight  m, 
are  dissolved  in  W  grams  of  solvent,  and  the  boiling-point  is  raised 


192 


THEORETICAL  CHEMISTRY 


A  degrees,  then,  since  m  grams  of  the  substance  when  dissolved 
in  100  grams  of  solvent,  produce  an  elevation  of  K  degrees  (the 
molecular  elevation),  it  follows  that 


1000. 


therefore, 


W 


:A  :: 


The  accompanying  table  gives  the  boiling-point  constants  for 
100  grams  and  100  cubic  centimeters  of  some  of  the  more  com- 
mon solvents. 


Solvent. 

Molecular  Elevation. 

100  gr. 

100  cc. 

Water  

5.2 

11.5 
21.0 
16.7 
26.7 
35.6 
30.1 

5.4 
15.6 
30.3 
22.2 
32.8 

Ethyl  alcohol  

Ether 

Acetone 

Benzene 

Chloroform 

Pyridine  

As  an  example  of  the  calculation  of  the  molecular  weight  of  a 
dissolved  substance  by  the  above  formula,  we  may  take  the 
calculation  of  the  molecular  weight  of  camphor  in  acetone  from 
the  following  data :  — 

When  0.674  gram  of  camphor  is  dissolved  in  6.81  grams  of 
acetone,  the  boiling-point  of  the  solvent  is  raised  1°.09.  Substitut- 
ing in  the  formula,  we  have 

0.674 


m  =  100  X  16.7  X 


6.81  X  1.09 


=  151. 


The  molecular  weight   of   camphor  according  to   the  formula 
CM  H160,  is  152. 

The  molecular  elevation  of  the  boiling-point  can  be  calculated 
by  means  of  the  formula, 

=  0.02  T2 
w 


DILUTE  SOLUTIONS  AND.  OSMOTIC  PRESSURE          193 


in  which  T  is  the  absolute  boiling-point  of  the  solvent,  and  w  is 
the  heat  of  vaporization  for  1  gram  of  the  solvent  at  its  boiling- 
point.  This  formula  will  be  derived  in  a  subsequent  paragraph 
of  this  chapter.  The  calculated  values 
of  K  are  in  close  agreement  with  the 
values  obtained  experimentally  by  Raoult 
and  others.  As  an  example,  the  calcu- 
lated value  of  the  molecular  elevation 
for  water,  the  heat  of  vaporization  of 
which  at  100°  C.  is  537  calories,  is 

0.02  X  (373)2         9 
-537- 

a  value  hi  exact  agreement  with  the  exper- 
imental value  given  in  the  table. 

Experimental  Determination  of  Mo- 
lecular Weight  by  the  Boiling-Point 
Method.  One  of  the  simplest  and  most 
convenient  of  the  various  forms  of  appa- 
ratus which  have  been  devised  for  the 
determination  of  the  boiling-point  of 
solutions,  is  that  developed  by  Jones,* 
and  shown  in  Fig.  63.  The  liquid  whose 
boiling-point  is  to  be  determined  is  intro- 
duced into  the  vessel  A,  which  already 
contains  a  platinum  cylinder  P,  em- 
bedded in  some  glass  beads.  Sufficient 
liquid  is  introduced  to  insure  the  com- 
plete covering  of  the  bulb  of  the  ther- 
mometer, as  shown  in  the  sketch. 

The  side  tube  of  A  is  connected  with  a  condenser,  C.  The 
vessel  A,  is  surrounded  by  a  thick  jacket  of  asbestos  J,  and  rests 
on  a  piece  of  asbestos  board  in  which  a  circular  hole  is  cut,  and 
over  which  a  piece  of  wire  gauze  is  placed.  The  liquid  is  heated 
by  means  of  a  burner,  B.  The  platinum  cylinder  is  the  feature 
which  differentiates  this  apparatus  from  the  various  other  forms 
[  *  Am.  Chem.  Jour.,  19,  581  (1897). 


Fig.  63. 


194  THEORETICAL  CHEMISTRY 

of  boiling-point  apparatus.  It  has  the  two-fold  object  of  pre- 
venting the  condensed  solvent  from  coming  in  direct  contact  with 
the  bulb  of  the  thermometer,  and  of  reducing  the  effect  of  radia- 
tion to  a  minimum.  The  liquid  in  A  is  boiled,  using  a  very  small 
flame,  until  the  thermometer  remains  constant;  this  temperature 
is  taken  as  the  boiling-point  of  the  liquid.  The  apparatus  is  now 
emptied  and  dried.  A  weighed  amount  of  the  liquid  is  then  intro- 
duced into  A,  and  to  this  is  added  a  known  weight  of  solute;  the 
thermometer  is  replaced  and  the  boiling-point  of  the  solution  is 
determined.  The  difference  between  the  readings  of  the  thermom- 
eter when  immersed  in  the  solution,  and  in  the  solvent  alone,  gives 
the  boiling  point  elevation.  For  further  details  concerning  the 
boiling-point  method  as  applied  to  the  determination  of  molecular 
weights,  the  student  is  referred  to  any  one  of  the  standard  labo- 
ratory manuals. 

Osmotic  Pressure  and  Boiling-Point  Elevation.  Imagine  a 
dilute  solution  containing  n  mols  of  solute  in  G  grams  of  solvent, 
and  let  d T  be  the  elevation  in  the  boiling-point.  Suppose  a  large 
quantity  of  the  solution  to  be  introduced  into  a  cylinder,  fitted 
with  a  frictionless  piston,  and  closed  at  the  bottom  by  a  semi- 
permeable  membrane.  Let  the  cylinder  and  contents  be  raised 
to  the  absolute  temperature  T°,  the  boiling-point  of  the  solu- 
tion, and  then  let  pressure  be  exerted  on  the  piston  just  sufficient 
to  overcome  the  osmotic  pressure  of  the  solution.  In  this  way,  let 
a  quantity  of  solvent  corresponding  to  1  mol  of  solute  be  forced 
through  the  semi-permeable  membrane.  The  volume  V,  thus 
expelled  is  the  volume  corresponding  to  G/n  grams  of  solvent. 
If  the  osmotic  pressure  of  the  solution  is  P,  then  the  work  done  in 
moving  the  piston  and  expelling  the  solvent  is  PV.  Now  let  the 
portion  of  the  solvent  which  has  been  forced  through  the  semi- 
permeable  membrane  be  vaporized.  For  this  operation  G/n  .  w 
calories  will  be  required,  w  being  the  heat  of  vaporization  for  1 
gram  of  solvent  at  its  boiling-point.  Then  let  the  entire  system 
be  raised  to  the  boiling-point  of  the  solution  (T  +  dT)°,  the  pre- 
viously expelled  G/n  grams  of  vapor  being  allowed  to  mix  with 
the  solution.  The  heat  of  vaporization  lost  at  T°  is  thus  recov- 
ered at  the  slightly  higher  temperature,  (T  +  dT)°.  Finally,  the 


DILUTE  SOLUTIONS   AND  OSMOTIC   PRESSURE          195 

entire  system  is  cooled  to  T°,  and  is  thus  restored  to  its  original 
state.  Applying  the  well-known  thermodynamic  relation,  that 
the  ratio  of  the  work  done  to  the  heat  absorbed,  is  the  same  as 
the  ratio  of  the  difference  in  temperature  to  the  absolute  initial 
temperature  of  the  system,  we  have 

PV       dT 


G          ~  T 

—  •  w 
n 


therefore, 


But,  since  PV  =  RT,  equation  (1)  may  be  written 


n 
"V  'G' 

If  n  =  1  and  G  =  100  grams,  then  dT  =  K,  (the  molecular  eleva- 
tion of  the  boiling-point),  or 

RT* 

100  w' 

Or  putting  R  =  2  calories,  we  have 

K=°™^.  (2) 

w 

Equation  (1)  shows  that  the  osmotic  pressure  of  a  solution  is  directly 
proportional  to  the  elevation  of  the  boiling-point.  Equation  (2)  was 
originally  derived  by  Van't  Hoff  at  about  the  time  when  Raoult 
determined  the  values  of  K  experimentally. 

Lowering  of  the  Freezing-Point.  Of  all  the  methods  employed 
for  the  determination  of  molecular  weights  in  solution,  the  freez- 
ing-point method  is  the  most  accurate  and  the  most  widely  used. 
It  was  pointed  out  by  Blagden  *  over  a  century  ago,  that  the  de- 
pression of  the  freezing-point  of  a  solvent  by  a  dissolved  substance  is 
directly  proportional  to  the  concentration  of  the  solution.  When 
equimolecular  quantities  of  different  substances  are  dissolved  in 
equal  volumes  of  the  same  solvent,  the  lowering  of  the  freezing- 
point  is  constant.  The  molecular  weight  of  any  soluble  sub- 
*  Phil.  Trans.,  78,  277  (1788). 


196 


THEORETICAL  CHEMISTRY 


stance  can  be  found,  as  in  the  boiling-point  method,  by  comparing 
its  effect  on  the  freezing-point  of  a  solvent  with  that  of  a  solute 
of  known  molecular  weight.  The  molecular  lowering  of  the  freezing- 
point,  or  the  freezing-point  constant,  of  a  solvent  is  defined  as  the  de- 
pression of  the  freezing-point  produced  by  dissolving  1  mol  of  solute 
in  100  grams  or  100  cubic  centimeters  of  solvent.  The  freezing-point 
constants  of  a  few  common  solvents  are  given  in  the  following  table. 


Solvent. 

Molecular  Depression. 

100  gr. 

100  cc. 

Water. 

18.5 
50 
74 
69 
39 

18.5 
56 

4i" 

Benzene  .  . 

Phenol  

Naphthalene  

Acetic  acid  

Van't  Hoff  showed  that  the  molecular  lowering  of  the  freezing- 
point  of  a  solvent  K,  can  be  calculated  from  the  absolute  freez- 
ing-point T,  and  the  heat  of  fusion  w,  for  1  gram  of  solvent  at 
the  temperature  T,  by  means  of  the  formula 

0.02  T2 


K  = 


w 


This  expression  is  analogous  to  that  which  applies  to  the  molec- 
ular elevation  of  the  boiling-point.  The  agreement  between  the 
observed  and  the  calculated  values  of  K  is  very  satisfactory,  as 
the  following  calculation  for  water  shows  :  — 

0.02  X  (273)2 

~80~  18'6' 


It  is  of  interest  to  note  that  the  calculated  value  of  K  for  water  is 
lower  than  the  experimental  values  originally  obtained  by  Raoult 
and  others.  Subsequent  experiments,  carried  out  with  greater 
care  and  better  apparatus,  by  Raoult,  Abegg  and  Loom  is  gave 
values  in  close  agreement  with  that  derived  theoretically.  A 
formula  analogous  to  that  employed  for  the  calculation  of  the 
molecular  weight  of  a  dissolved  substance  from  the  elevation  it 


DILUTE  SOLUTIONS  AND  OSMOTIC   PRESSURE          197 

produces  in  the  boiling-point  of  a  solvent,  may  be  used  for  the 
calculation  of  molecular  weight  from  freezing-point  depression. 
Thus,  if  g  grams  of  solute  when  dissolved  in  W  grams  of  solvent 
produce  a  depression  A  of  the  freezing-point  of  the  solvent,  the 
molecular  weight  m,  is  given  by  the  formula 


where  K  is  the  molecular  lowering  of  the  freezing-point. 

EXAMPLE.  When  1.458  grams  of  acetone  are  dissolved  in 
100  grams  of  benzene,  the  freezing-point  of  the  solvent  is  de- 
pressed 1.22°,  therefore  the  molecular  weight  of  acetone  is 


The  molecular  weight  of  acetone,  calculated  from  the  formula 
C3H6O,  is  58. 

In  order  to  obtain  trustworthy  results  with  the  freezing-point 
method,  it  is  necessary  that  only  the  pure  solvent  separate  out 
when  the  solution  freezes,  and  that  excessive  overcooling  be 
avoided.  When  too  great  overcooling  occurs,  the  subsequent 
freezing  of  the  solution  results  in  the  separation  of  so  large  an 
amount  of  solvent  in  the  solid  state,  that  the  observed  freezing 
point  corresponds  to  the  equilibrium  temperature  of  a  more 
concentrated  solution  than  that  originally  prepared.  A  formula 
for  the  correction  of  the  concentration,  due  to  excessive  overcooling, 
has  been  derived  by  Jones.*  If  the  overcooling  of  the  solution 
in  degrees  be  represented  by  u,  the  heat  of  fusion  of  1  gram  of  sol- 
vent at  the  freezing-point  by  w,  and  the  specific  heat  of  the  sol- 
vent by  c,  then  the  fraction  of  the  solvent  which  will  solidify,  /, 
may  be  calculated  by  the  formula, 

f  =  —  . 
w' 

When  water  is  used  as  the  solvent,  c  =  1  and  w  =  80.  There- 
fore, for  every  degree  of  overcooling,  the  fraction  of  the  solvent 
separating  as  ice  will  be  1/80,  and  the  concentration  of  the  original 

*  Zeit.  phys.  Chem.,  12,  624  (1893). 


198 


THEORETICAL  CHEMISTRY 


solution  is  increased  by  just  so  much.  It  is  simpler,  however,  to 
apply  the  correction  directly  to  the  freezing-point  depression  in- 
stead of  to  the  concentration. 

Experimental  Determination  of  Molecular  Weight  by  the 
Freezing-Point  Method.  The  apparatus  almost  universally  em- 
ployed for  the  determination  of  molecular  weights  by  the  freezing- 
point  method  is  that  devised  by  Beckmann,* 
and  shown  in  Fig.  64.  It  consists  of  a  thick- 
walled  test  tube  A,  provided  with  a  side  tube, 
and  fitted  into  a  wider  tube  A  i,  thus  surround- 
ing A  with  an  air  space. 

The  whole  is  fitted  into  the  metal  cover  of 
a  large  battery  jar,  which  is  filled  with  a  freez- 
ing mixture  whose  temperature  is  several  de- 
grees below  the  freezing-point  of  the  solvent. 

The  tube  A  is  closed  by  a  cork  stopper, 
through  which  passes  the  thermometer  and 
stirrer.  The  thermometer  is  generally  of  the 
Beckmann  differential  type.  This  instrument 
has  a  scale  about  6°  in  length,  each  degree 
being  divided  into  hundredths;  the  quantity 
of  mercury  in  the  bulb  can  be  varied  by  means 
of  a  small  reservoir  at  the  top  of  the  scale,  so 
that  the  zero  of  the  instrument  can  be  adjusted 
for  use  with  solvents  of  widely  different  freez- 
ing-points. In  carrying  out  a  determination 
with  the  Beckmann  apparatus,  a  weighed 


Fig.  64. 


quantity  of  solvent  is  placed  in  A,  and  the  temperature  of  the 
refrigerating  mixture  regulated  so  as  to  be  not  more  than  5°  below 
the  freezing-point  of  the  solvent.  The  tube  A  is  removed  from  its 
jacket,  and  is  immersed  in  the  freezing  mixture  until  the  solvent 
begins  to  freeze.  It  is  then  replaced  in  the  jacket  A\t  and  the 
solvent  is  vigorously  stirred.  The  thermometer  rises  during  the 
stirring  until  the  true  freezing-point  is  reached,  after  which  it 
remains  constant.  This  temperature  is  taken  as  the  freezing 
temperature  of  the  solvent. 

*  Zeit.  phys.  Chem.,  2,  683  (1888) . 


DILUTE  SOLUTIONS  AND  OSMOTIC  PRESSURE          199 

The  tube  A  is  now  removed  from  the  freezing  mixture,  and  a 
weighed  amount  of  the  substance  whose  molecular  weight  is  to 
be  determined  is  introduced.  When  the  substance  has  dissolved, 
the  tube  is  replaced  in  AI  and  the  solution  cooled  not  more  than 
a  degree  below  its  freezing-point.  A  small  fragment  of  the  solid 
solvent  is  dropped  into  the  solution,  which  is  then  stirred 
vigorously  until  the  thermometer  remains  constant.  The  maxi- 
mum temperature  is  taken  as  the  freezing-point  of  the  solution. 
The  difference  between  the  freezing-points  of  solution  and 
solvent  is  the  depression  sought.  For  further  details  con- 
cerning the  determination  of  molecular  weights  by  the  freezing- 
point  method,  the  student  is  referred  to  a  physico-chemical 
laboratory  manual. 

Osmotic  Pressure  and  Freezing-Point  Depression.  Let  dT 
be  the  freezing-point  depression  produced  by  n  mols  of  solute  in 
G  grams  of  solvent,  the  solution  being  dilute.  Imagine  a  large 
quantity  of  this  solution  to  be  confined  within  a  cylinder  fitted 
with  a  frictionless  piston,  the  bottom  of  the  cylinder  being  closed 
by  a  semi-permeable  membrane.  Let  the  cylinder  and  contents 
be  cooled  to  the  freezing  temperature  of  the  solvent  T,  and  then 
let  pressure  be  applied  to  the  piston  until  a  quantity  of  solvent 
corresponding  to  1  mol  of  solute  is  forced  through  the  semi-perme- 
able membrane.  This  requires  an  expenditure  of  energy  equiva- 
lent to  PV,  where  P  is  the  osmotic  pressure  of  the  solution  and  V 
is  the  volume  of  solvent  expelled.  The  volume  V  is  clearly  the 
volume  of  G/n  grams  of  solvent.  Now  let  the  expelled  portion 

ry 

of  solvent  be  frozen  and  the  system  deprived  of  —  w  calories  of 

heat,  where  w  is  the  heat  of  fusion  of  1  gram  of  the  solvent  at  the 
temperature  T. 

The  temperature  of  the  solution  is  then  lowered  to  its  freezing- 
point  (T  —  dT),  and  the  G/n  grams  of  solidified  solvent  dropped 
into  it.  The  solidified  solvent  melts,  thereby  restoring  to  the 
system  at  the  temperature  (T  —  dT),  the  heat  of  fusion  formerly 
taken  from  it.  Finally,  the  temperature  of  the  system  is  raised 
to  T,  the  initial  temperature  of  the  cycle.  Applying  the  familiar 
thermodynamic  relation,  that  the  ratio  of  the  work  done  to  the 


200  THEORETICAL  CHEMISTRY 

heat  absorbed,  is  the  same  as  the  ratio  of  the  difference  in  temper- 
ature to  the  initial  absolute  temperature,  we  have 


From  which  we  obtain 


.  a) 

„  ^ 

n 


w      G 
But  PV  —  RT,  hence  equation  (1)  becomes 


W       Or 

If  n  =  1  and  G  =  100  grams,  then  dT  =  K,  the  molecular  lower- 
ing of  the  freezing-point,  and 

RT* 

100  W' 

Or  putting  R  =  2  calories,  we  have 

K  =  5^I_2.  (2) 

An  equation  to  which  reference  has  already  been  made. 

It  is  evident  from  equation  (1)  that  the  osmotic  pressure  of  a 
solution  is  directly  proportional  to  the  freezing  point  depression. 

Molecular  Weight  in  Solution.  As  has  been  pointed  out,  the 
molecular  weight  of  a  dissolved  substance  can  be  readily  calcu- 
lated, provided  that  the  osmotic  pressure  of  a  dilute  solution  of 
known  concentration  at  known  temperature  is  determined.  But 
the  experimental  difficulties  attending  the  direct  measurement 
of  osmotic  pressure  are  so  great,  that  it  is  customary  to  employ 
other  methods  based  upon  properties  of  dilute  solutions  which 
are  proportional  to  osmotic  pressure.  We  have  shown  that  in 
dilute  solutions  osmotic  pressure  is  directly  proportional  (1)  to 
the  relative  lowering  of  the  vapor  pressure,  (2)  to  the  elevation 
of  the  boiling-point,  and  (3)  to  the  depression  of  the  freezing-point. 

From  this  it  follows,  that  equimolecular  quantities  of  different 
substances  dissolved  in  equal  volumes  of  the  same  solvent,  exert  the 


DILUTE  SOLUTIONS  AND  OSMOTIC  PRESSURE         201 

same  osmotic  pressure,  and  produce  the  same  relative  lowering  of 
vapor  pressure,  the  same  elevation  of  boiling-point,  and  the  same 
depression  of  freezing-point.  Since  equimolecular  quantities  of 
different  substances  contain  the  same  number  of  molecules,  it  is 
evident  that  the  magnitude  of  osmotic  pressure,  relative  lowering  of 
vapor  pressure,  elevation  of  boiling-point  and  depression  of  freezing- 
point,  is  dependent  upon  the  number  of  particles  present  in  the  solu- 
tion and  is  independent  of  their  nature.  It  has  been  pointed  out 
by  Nernst  that  any  process  which  involves  the  separation  of 
solvent  from  solute,  may  be  employed  for  the  determination  of 
molecular  weights.  A  little  reflection  will  convince  the  reader 
that  the  four  methods  discussed  in  this  chapter  involve  such  separ- 
ation. Both  Van't  Hoff  and  Raoult  emphasized  the  fact  that  the 
formulas  derived  for  the  determination  of  molecular  weights  in  so- 
lution depend  upon  assumptions  which  are  valid  only  for  dilute 
solutions.  It  follows,  therefore,  that  we  are  not  justified  in  apply- 
ing these  formulas  to  concentrated  solutions.  Up  to  the  present  time 
we  have  no  satisfactory  theory  of  concentrated  solutions,  neither 
can  we  state  up  to  what  concentration  the  gas  laws  apply. 

PROBLEMS. 

1.  At  10°  C.  the  osmotic  pressure  of  a  solution  of  urea  is  500  mm. 
of  mercury.     If  the  solution  is  diluted  to  ten  times  its  original  volume, 
what  is  the  osmotic  pressure  at  15°  C.?  Ans.   50.89  mm. 

2.  The  osmotic  pressure  of  a  solution  of  0.184  gram  of  urea  in  100  cc. 
of  water  was  56  cm.  of  mercury  at  30°  C.     Calculate  the  molecular  weight 
of  urea.  Ans.   62.03. 

3.  At  24°  C.  the  osmotic  pressure  of  a  cane  sugar  solution  is  2.51  atmos- 
pheres.   What  is  the  concentration  of  the  solution  in  mols  per  liter? 

Ans.  0.103. 

4.  At  25°.  1  C.  the  osmotic  pressure  of  solution  of  glucose  containing 
18  grams  per  liter  was  2.43  atmospheres.     Calculate  the  numerical  value 
of  the  constant  R,  when  the  unit  of  energy  is  the  gram-centimeter. 

Ans.  84,300. 

5.  The  vapor  pressure  of  ether  at  20°  C.  is  442  mm.  and  that  of  a  solu- 
tion of  6.1  grams  of  benzoic  acid  in  50  grams  of  ether  is  410  mm.  at  the 
same  temperature.    Calculate  the  molecular  weight  of  benzoic  acid  in 
ether.  Ans.   124. 


202  THEORETICAL  CHEMISTRY 

6.  At  10°  C.  the  vapor  pressure  of  ether  is  291.8  mm.  and  that  of  a 
solution  containing  5.3  grams  of  benzaldehyde  in  50  grams  of  ether  is 
271.8  mm.    What  is  the  molecular  weight  of  benzaldehyde? 

Ans.   109. 

7.  A  solution  containing  0.5042  gram  of  a  substance  dissolved  in  42.02 
grams  of  benzene  boils  at  80°.  175  C.     Find  the  molecular  weight  of  the 
solute,  having  given  that  the  boiling-point  of  benzene  is  80°. 00  C.,  and  its 
heat  of  vaporization  is  94  calories  per  gram.  Ans.   181.9. 

8.  A  solution  containing  0.7269  gram  of  camphor  (mol.  wt.  =  152) 
in  32.08  grams  of  acetone  (boiling-point  =  56°.30  C.)  boiled  at  56°.55  C. 
What  is  the  molecular  elevation  of  the  boiling-point  of  acetone?    What  is 
its  heat  of  vaporization? 

Ans.  K  =  16.74;  w  =  129.5  cals.  per  gm. 

9.  A  solution  of  9.472  grams  of  CdI2  in  44.69  grams  of  water  boiled  at 
100°.303  C.    The  heat  of  vaporization  of  water  is  536  calories  per  gram. 
What  is  the  molecular  weight  of  CdI2  in  the  solution?    What  conclusion 
as  to  the  state  of  CdI2  in  solution  may  be  drawn  from  the  result? 

Ans.  *63.2. 

10.  The  freezing-point  of  pure  benzene  is  5°.440  C.  and  that  of  a  solu- 
tion containing  2.093  grams  of  benzaldehyde  in  100  grams  of  benzene  is 
4°.440  C.     Calculate  the  molecular  weight  of  benzaldehyde  in  the  solu- 
tion.   K  for  benzene  is  50.  Ans.'  104.6. 

11.  A  solution  of  0.502  gram  of  acetone  in  100  grams  of  glacial  acetic 
acid  gave  a  depression  of  the  freezing-point  of  0°.339  C.     Calculate  the 
molecular  depression  for  glacial  acetic  acid.  Ans.   39. 

12.  By  dissolving  0.0821  gram  of  m-hydroxybenzaldehyde  (C7H602)  in 
20  grams  of  naphthalene  (melting  point  80°.  1  C.)  the  freezing-point  is 
lowered  by  0°.232  C.    Assuming  that  the  molecular  weight  of  the  solute 
is  normal  in  the  solution,  calculate  the  molecular  depression  for  naphtha- 
lene and  the  heat  of  fusion  per  gram. 

Ans.  K  =  68.96;    w  =  36.2  cals.  per  gm. 


CHAPTER  IX. 
ASSOCIATION,  DISSOCIATION  AND   SOLVATION. 

Abnormal  Solutes.  As  has  already  been  pointed  out  the 
acceptance  of  Avogadro's  hypothesis  was  greatly  retarded  by  the 
discovery  of  certain  substances  whose  vapor  densities  were  ab- 
normal. Thus,  the  vapor  density  of  ammonium  chloride  is  approx- 
imately one-half  of  that  required  by  the  formula  NH4C1,  while 
the  vapor  density  of  acetic  acid  corresponds  to  a  formula  whose 
molecular  weight  is  greater  than  that  calculated  from  the  formula, 
C2H402.  The  anomalous  behavior  of  ammonium  chloride  and 
kindred  substances  has  been  shown  to  be  due,  not  to  a  failure  of 
Avogadro's  law,  but  to  a  breaking  down  of  the  molecules,  —  a 
process  known  as  dissociation.  The  abnormally  large  molecular 
weight  of  acetic  acid  on  the  other  hand,  has  been  ascribed  to  a 
process  of  aggregation  of  the  normal  molecules,  known  as  asso- 
ciation. In  extending  the  gas  laws  to  dilute  solutions  similar 
phenomena  have  been  encountered. 

Association  in  Solution.  When  the  molecular  weight  of  acetic 
acid  in  benzene  is  determined  by  the  freezing-point  method,  the 
depression  of  the  freezing-point  is  abnormally  small  and  conse- 
quently, as  the  formula 


shows,  the  molecular  weight  will  be  greater  than  that  .correspond- 
ing to  the  formula,  C2H4O2.  Acetic  acid  in  benzene  solution  is 
thus  shown  to  be  associated.  Almost  all  compounds  containing 
the  hydroxyl  and  cyanogen  groups  when  dissolved  in  benzene  are 
found  to  be  associated.  Solvents,  such  as  benzene  and  chloro- 
form, are  frequently  termed  associating  solvents,  although  it  is 
doubtful  whether  they  exert  any  associating  action.  There  is 
considerable  experimental  evidence  to  show  that  those  substances 

203 


204  THEORETICAL  CHEMISTRY 

whose  molecules  arc  associated  in  benzene  and  chloroform  solu- 
tion, are  also  associated  in  the  free  condition.  Just  as  the  depres- 
sion of  the  freezing-point  of  a  solution  of  an  associated  substance 
is  abnormally  small,  so  its  osmotic  pressure  and  other  related 
properties  will  be  less  than  the  calculated  values. 

Dissociation  in  Solution.  Van't  Hoff  *  pointed  out  that  the 
osmotic  pressure  of  solutions  of  most  salts,  of  all  strong  acids,  and 
of  all  strong  bases  is  much  greater  for  all  concentrations  than 
would  be  expected  from  the  osmotic  pressure  of  solutions  of  sub- 
stances, like  cane  sugar  or  urea,  for  corresponding  concentrations. 
He  was  unable  to  account  for  this  abnormal  behavior,  and  in  order 
to  render  the  general  gas  equation  applicable,  he  introduced  a 
factor  i,  the  modified  equation  being 

PV  =  iRT. 

If  the  osmotic  pressure  of  some  substance,  like  cane  sugar,  which 
behaves  normally,  be  represented  by  P0,  the  factor  i  is  given  by 
the  expression 

;=-• 

Since  the  osmotic  pressure  of  a  solution  is  proportional  to  the 
relative  lowering  of  its  vapor  pressure,  to  the  elevation  of  its 
boiling-point,  and  to  the  lowering  of  its  freezing-point,  we  may 
write 


o       Pi  ~  Po          o 

Pi 

where  the  symbols  have  their  usual  significance.  The  subscript 
0  refers  in  each  case  to  a  substance  which  behaves  normally,  and  A 
denotes  either  boiling-point  elevation  or  freezing-point  depression. 
A  more  definite  conception  of  the  abnormal  behavior  of  salts 
will  be  gained  by  an  inspection  of  the  accompanying  tables.  In 
the  first  column  is  recorded  the  molar  concentration  of  the  solu- 
tion; the  second  column  gives  the  observed  depressions  of  the 

*  Zeit.  phys.  Chem.,  i,  501  (1887). 


ASSOCIATION,  DISSOCIATION  AND  SOLVATION         205 


freezing-point  and  the  third  column  contains  the  values  of  the 
ratio  of  the  observed  depression  to  the  normal  depression,  or  i. 


POTASSIUM  CHLORIDE. 


POTASSIUM  SULPHATE. 


m 

A 

t 

TO 

A 

I 

0.05 

0.1750 

1.88 

0.05 

0.2270 

2.33 

0.10 

0.3445 

1.85 

0.10 

0.4317 

2.32 

0.20 

0.6808 

1.83 

0.20 

0.8134 

2.18 

0.40 

1.3412 

1.80 

0.30 

1.1673 

2.09 

ALUMINIUM  CHLORIDE. 


SODIUM  CHROMATE. 


m 

!• 

i 

m 

A 

i 

0.046 
0.076 
0.102 

0.276 
0.446 
0.578 

3.22 
3.15 
3.04 

0.1 
0.2 
0.5 
1  0 

0.450 
0.850 
1.960 
3  800 

2.42 
2.28 
2.11 
2  04 

It  is  apparent  from  the  above  data  that  the  depression  of  the 
freezing-point  of  water  caused  by  these  salts  is  abnormally  large, 
a  fact  which  points  to  an  increase  in  the  number  of  dissolved  units 
over  that  corresponding  to  the  initial  concentration. 

The  Theory  of  Electrolytic  Dissociation.  In  1887,  Arrhenius  * 
advanced  an  hypothesis  to  account  for  the  abnormal  osmotic 
activity  of  solutions  of  acids,  bases  and  salts.  He  pointed  out 
that  just  as  the  exceptional  behavior  of  certain  gases  has  been 
completely  reconciled  with  the  law  of  Avogadro,  by  assuming  a 
dissociation  of  the  vaporized  molecule  into  two  or  more  simpler 
molecules,  so  the  enhanced  osmotic  pressure  and  the  abnormally 
great  freezing-point  depression  of  solutions  of  acids,  bases  and 
salts  can  be  explained,  if  we  assume  a  similar  process  of  dissoci- 
ation. He  proposed,  therefore,  that  aqueous  solutions  of  acids, 
bases  and  salts  be  considered  as  dissociated,  to  a  greater  or  less 
extent,  into  positively-  and  negatively-charged  particles  or  ions, 
and  that  the  increase  in  the  number  of  dissolved  units  due  to  this 

*  Zeit.  phys.  Chem.,  i,  631  (1887). 


206  THEORETICAL  CHEMISTRY 

dissociation  is  the  cause  of  the  enhanced  osmotic  activity.  Accord- 
ing to  this  hypothesis,  hydrochloric  acid,  potassium  hydroxide  and 
potassium  chloride,  when  dissolved  in  water,  dissociate  hi  the 
following  manner:  — 

HC1    ->H*  +  C1' 


KC1   -»K'  +  Cr, 

where  the  dots  indicate  positively-charged  ions  and  the  dashes 
negatively-charged  ions. 

In  each  of  the  above  cases,  one  molecule  yields  two  ions,  so 
that,  if  dissociation  is  complete,  the  maximum  osmotic  effect 
should  not  be  greater  than  twice  that  produced  by  an  equimolecu- 
lar  quantity  of  a  substance  which  behaves  normally.  Reference 
to  the  preceding  table  shows  that  the  value  of  i  for  potassium 
chloride  approaches  the  limiting  value  of  2  as  the  solution  is 
diluted.  The  other  salts  given  in  the  table  dissociate,  according 
to  Arrhenius,  in  the  following  way  :  — 


Na*Cr04-»2Na+Cr04' 


If  these  equations  correctly  represent  the  process  of  dissociation, 
then  when  dissociation  is  complete,  the  osmotic  effect  of  infinitely 
dilute  solutions  of  potassium  sulphate  and  sodium  chromate 
should  be  three  times  the  effect  produced  by  an  equimolecular 
quantity  of  a  normal  solute,  and  in  the  case  of  aluminium  chloride, 
the  maximum  effect  should  be  four  times  the  effect  due  to  a  normal 
substance.  A  glance  at  the  table  shows  that  the  values  of  i  for 
solutions  of  the  three  salts  approach  these  respective  limits.  If 
this  hypothesis  of  ionic  dissociation  be  accepted,  then  it  becomes 
possible  to  calculate  the  degree  of  ionization  in  any  solution  by 
comparing  its  freezing-point  depression  with  the  freezing-point 
lowering  of  an  equimolecular  solution  of  a  normal  substance. 

Let  us  suppose  that  the  degree  of  dissociation  of  1  molecule  of 
a  dissolved  substance  is  a,  each  molecule  yielding  n  ions.    Then 


ASSOCIATION,  DISSOCIATION  AND  SOLVATION          207 


1  —  a  will  be  the  undissociated  portion  of  the  molecule,  and  the 
total  number  of  dissolved  units  will  be 

1  —  a 


na. 


If  A  is  the  depression  of  the  freezing-point  produced  by  the  sub- 
stance, and   AO  the   depression   produced  by  an   equimolecular 
quantity  of  an  undissociated  substance,  then 
1  —  a  +  na      A 


1 


or 


a  = 


i-  I 
n-1 


It  will  be  observed  that  this  formula  is  identical  with  that  derived 
for  the  degree  of  dissociation  of  a  gas  (p.  66).  If  this  formula 
be  applied  to  the  freezing-point  data  for  solutions  of  potassium 
chloride  given  in  the  preceding  table  we  find  the  following  per- 
centages of  dissociation  corresponding  to  the  different  concentra- 
tions :  — 

POTASSIUM  CHLORIDE. 


m 

A 

t 

a 

0.05 

0.1750 

1.88 

% 

88 

0.10 

0.3445 

1.85 

85 

0.20 

0.6808 

1.83 

83 

0.40 

1.3412 

1.80 

80 

The  figures  in  the  last  column  show  that  the  degree  of  dissocia- 
tion increases  as  the  concentration  diminishes.  It  was  further 
pointed  out  by  Arrhenius  that  all  of  the  substances  which  exhibit 
abnormal  osmotic  effects,  when  dissolved  in  water,  yield  solutions 
which  conduct  the  electric  current,  whereas,  aqueous  solutions  of 
such  substances  as  cane  sugar,  urea  and  alcohol,  exert  normal 
osmotic  pressures,  but  do  not  conduct  electricity  any  better  than 
the  pure  solvent.  In  other  words,  only  electrolytes  are  capable  of 
undergoing  ionic  dissociation;  hence  Arrhenius  termed  his  hypo- 
thesis the  electrolytic  dissociation  theory.  As  we  have  seen,  when 
potassium  chloride  is  dissolved  in  water,  it  is  supposed  to  dissoci- 


208 


THEORETICAL  CHEMISTRY 


ate  into  positively-charged  potassium  ions  and  negatively-charged 
chlorine  ions.  Accordingly  when  two  platinum  electrodes,  one 
charged  positively  and  the  other  negatively,  are  introduced  into 
the  solution,  the  potassium  ions  move  toward  the  negative  elec- 
trode and  the  chlorine  ions  move  toward  the  positive  electrode,  the 
passage  of  a  current  through  the  solution  consisting  in  the  ionic 
transfer  of  electric  charges.  Since  the  undissociated  molecules, 
being  electrically  neutral,  do  not  participate  in  the  transfer  of 
electric  charges,  it  follows  that  the  conductance  of  a  solution  of 
an  electrolyte  is  dependent  upon  the  degree  of  dissociation.  The 
relation  between  electrical  conductance  and  the  degree  of  ioniza- 
tion  will  be  discussed  in  a  subsequent  chapter.  It  may  be  stated 
at  this  point,  however,  that  the  values  of  a  based  upon  measure- 
ments of  electrical  conductance,  while  showing  some  discrepancies 
in  individual  cases,  are  in  general  in  good  agreement  with  the 
values  obtained  by  the  freezing-point  method.  Furthermore,  the 
values  of  a  obtained  from  freezing-point  measurements  are  in  har- 
mony with  those  based  upon  De  Vries'  isotonic  coefficients. 
It  will  be  seen,  on  referring  to  the  table  of  isotonic  coefficients 
(p.  181),  that  solutions  of  electrolytes  show  enhanced  osmotic 
activity.  Thus,  the  osmotic  pressures  of  equi-molecular  solutions 
of  cane  sugar,  potassium  nitrate  and  calcium  chloride  are  to  each 
other  as  1  :  1.67  :  2.40. 

The  following  table  illustrates  the  general  agreement  between 
the  values  of  i  calculated,  (a)  from  electrical  conductance,  (b) 
from  freezing-point  depression,  and  (c)  from  De  Vries'  isotonic 
coefficients. 


Substance. 

Molar  Cone. 

(a) 

(b) 

(c) 

KC1 

0  14 

1  86 

1  82 

1  81 

LiCl  

0  13 

1  84 

1  94 

1  92 

Ca(NO3)2.. 

0  18 

2  46 

2  47 

2  48 

MgCl2.. 

0.19 

2.48 

2  68 

2  79 

CaCU 

0  184 

2  42 

2  67 

2  78 

It  must  be  remembered  that  the  values  of  i  derived  from  freez- 
ing-point   measurements    correspond    to    temperatures    in    the 


ASSOCIATION,   DISSOCIATION  AND  SOLVATION          209 

neighborhood  of  0°  C.,  while  those  derived  from  the  other 
methods  correspond  to  temperatures  ranging  from  17°  C.  to 
25°  C. 

Chemical  Properties  of  Completely  Ionized  Solutions.  The 
chemical  properties  of  an  ion  are  very  different  from  the  properties 
of  the  atom  or  radical  when  deprived  of  its  electrical  charge. 
For  example,  the  sodium  ion  is  present  in  an  aqueous  solution  of 
sodium  chloride,  but  there  is  no  evidence  of  chemical  reaction  with 
the  solvent;  whereas,  the  element  in  the  electrically-neutral  con- 
dition reacts  violently  with  water,  evolving  hydrogen  and  form- 
ing a  solution  of  potassium  hydroxide.  Again,  take  the  element 
chlorine:  when  chlorine  in  the  molecular  condition,  either  as  gas 
or  in  solution,  is  added  to  a  solution  of  silver  nitrate,  no  precipitate 
of  silver  chloride  is  formed.  Further,  chlorine  in  such  compounds 
as  CHC13,  CC14,  etc.,  is  not  precipitated  by  silver  nitrate,  since 
these  compounds  are  not  dissociated  by  water.  Or,  chlorine  may 
be  present  in  a  compound  which  is  dissociated  by  water  and  yet 
not  exhibit  its  characteristic  reactions,  because  it  is  present  in  a 
complex  ion. 

Thus,  potassium  chlorate  dissociates  in  the  following  manner:  — 

KC1O3  -»  K*  +  CKY. 

On  adding  silver  nitrate,  there  is  no  precipitation,  because  the 
chlorine  forms  a  complex  ion  with  oxygen. 

Physical  Properties  of  Completely  Ionized  Solutions.  The 
physical  properties  of  completely  ionized  solutions  are,  in  general, 
additive.  This  is  well  illustrated  by  a  series  of  solutions  of 
colored  salts,  the  color  of  which  is  due  to  the  presence  of  a  par- 
ticular ion.  It  is  found,  when  the  solutions  are  sufficiently 
dilute  to  insure  complete  dissociation,  that  they  all  have  the 
same  color.  The  additive  character  of  the  colors  of  solutions  of 
electrolytes  is  brought  out  in  a  striking  manner  by  a  comparison 
of  their  absorption  spectra.  Ostwald  *  photographed  the  absorp- 
tion spectra  of  solutions  of  the  permanganates  of  lithium,  cadmium, 
ammonium,  zinc,  potassium,  nickel,  magnesium,  copper,  hydrogen 
and  aluminium,  each  solution  containing  0.002  gram-equivalents 

*  Zeit.  phys.  Chera.,  9,  579  (1892). 


210  THEORETICAL  CHEMISTRY 

of  salt  per  liter.  The  absorption  spectra,  as  shown  in  Fig.  65, 
will  be  seen  to  be  practically  identical,  the  bands  occupying  the 
same  position  in  each  spectrum.  This  affords  a  strong  con- 
firmation of  the  theory  of  electrolytic  dissociation,  according  to 

which  a  dilute  solution  is  to  be 
regarded  as  a  mixture  of  elec- 
trically equivalent  quantities  of 
oppositely  charged  ions,  each  of 
which  contributes  its  specific 
properties  to  the  solution.  The 
permanganate  ion  being  colored, 
and  common  to  all  of  the  salts 
examined,  and  the  positive  ions 
of  the  various  substances  being 
colorless,  it  follows  that  when  dis- 
sociation is  complete,  the  absorp- 
tion spectra  of  all  of  the  solutions 
must  be  identical.  A  number  of 
other  properties  of  completely 
dissociated  solutions  have  been 
shown  to  be  additive.  Among 
these  may  be  mentioned  density, 
specific  refraction,  surface  ten- 
sion, thermal  expansion,  and 
magnetic  rotatory  power.  Addi- 
tional evidence  in  favor  of  the 
theory  of  electrolytic  dissociation 
will  be  furnished  in  forthcoming 
chapters.  Notwithstanding  the 
large  number  of  facts  which  can 

be  satisfactorily  interpreted  by  the  theory,  there  are  directions 
in  which  it  requires  amplification  and  modification.  Of  the 
various  objections  which  have  been  urged  against  the  theory  of 
electrolytic  dissociation,  one  is  of  sufficient  weight  to  call  for 
brief  consideration  here.  When  two  elements,  such  as  potassium 
and  chlorine,  combine  to  form  potassium  chloride,  the  reaction 
is  violent  and  a  large  amount  of  heat  is  developed.  Nevertheless, 


ASSOCIATION,  DISSOCIATION  AND  SOLVATION         211 

according  to  the  ionization  theory,  the  strong,  mutual  affinity  of 
these  two  elements  is  overcome  by  the  act  of  solution  in  water, 
the  molecule  being  split  into  two  oppositely-charged  atoms.  Obvi- 
ously such  a  separation  calls  for  the  expenditure  of  a  large  amount 
of  energy,  and  the  question  naturally  arises :  —  What  is  the 
source  of  this  energy?  While  this  question  cannot  be  fully  an- 
swered here,  it  may  be  pointed  out  that  we  have  abundant 
evidence  to  show  that  the  ions  are  hydrated,  each  being  surrounded 
by  an  " atmosphere"  of  solvent.  In  view  of  this  fact,  it  has  been 
suggested  *  that  dissociation  in  aqueous  solution  is  caused  by  the 
mutual  attraction  between  the  ions  and  the  molecules  of  the  sol- 
vent, the  heat  of  ionic  hydration  furnishing  the  energy  necessary 
for  the  separation  of  the  ions. 

Freezmg-Point  Depressions  Produced  by  Concentrated  Solu- 
tions of  Electrolytes.  As  has  already  been  mentioned,  the 
dissociation  of  electrolytes  in  aqueous  solution  increases  with  the 
dilution,  becoming  complete  at  a  concentration  of  about  0.001 
molar.  We  should  expect  the  dissociation  to  diminish  with  in- 
creasing concentration,  until,  if  the  electrolyte  is  sufficiently  solu- 
ble, the  depression  of  the  freezing-point  becomes  normal.  Recent 
investigations  by  Jones  and  his  co-workers  f  have  shown  that  the 
facts  are  contradictory  to  this  expectation.  They  found  that  the 
value  of  the  molecular  depression  of  the  freezing-point  of  water 
produced  by  a  number  of  chlorides  and  bromides,  diminished  with 
increasing  concentration  up  to  a  certain  point,  as  would  be  ex- 
pected, and  then  increased  again.  The  increase  in  the  molecular 
depression  became  very  marked  at  great  concentrations;  in  fact, 
the  molecular  depression  in  a  molar  solution  was  frequently  greater 
than  the  molecular  depression  corresponding  to  a  completely 
dissociated  salt.  This  phenomenon  was  systematically  studied 
by  Jones  and  the  author  t  and  the  fact  was  established  that  it 
is  quite  general. 

*  Trans.  Faraday  Soc.,  i,  197  (1905);  3,  123  (1907). 

t  Am.  Chem.  Jour.,  22,  5,  110  (1899);  23,  89  (1900). 

t  Zeit.phys.  Chem.,  46,  244  (1903);  Phys.Rev.,  18,  146  (1904);  Am.  Chem. 
Jour.,  31,  303  (1904);  32,  308  (1904);  33,  534  (1905);  34,  291  (1905);  Zeit. 
phys.  Chem.,  49,  385  (1904);  Monograph  No.  60,  Carnegie  Institution  of 
Washington. 


212 


THEORETICAL  CHEMISTRY 


The  accompanying  tables  give  the  results  obtained  by  Jones 
and  Getman  for  a  few  typical  solutions;  m  is  the  molar  con- 
centration, A  is  the  observed  depression  of  the  freezing-point, 
and  A/ra  is  the  molecular  depression. 

CALCIUM  CHLORIDE. 


m 

A 

A/m 

0.102 

0.505° 

4.98 

0.153 

0.752 

4.91 

0.204 

1.012 

4.96 

0.255 

1.267 

4.97 

0.306 

1.537 

5.02 

0.408 

2.104 

5.16 

0.510 

2.681 

5.26 

0.612 

3.348 

5.47 

1.000 

6.345 

6.345 

1.500 

11.296 

7.531 

2.000 

17.867 

8.934 

1.949 

17.710 

9.03 

2.274 

23.000 

10.11 

2.598 

29.000 

11.16 

2.923 

37.400 

12.79 

3.248 

46.500 

14.32 

MAGNESIUM  BROMIDE. 


m 

A 

A/m 

0.0517 

0.277° 

5.36 

0.103 

0.531 

5.14 

0.155 

0.801 

5.17 

0.207 

1.088 

5.26 

0.310 

1.690 

5.45 

0.414 

2.347 

5.67 

0.517 

3.022 

5.84 

0.321 

1.691 

5.27 

0.642 

3.921 

6.17 

0.964 

6.850 

7.11 

1.610 

15.200 

9.44 

2.571 

37.500 

14.60 

ASSOCIATION,  DISSOCIATION  AND  SOLVATION 
ALUMINIUM  CHLORIDE. 


213 


m 

A 

A/m 

0.046 

0.276° 

6.04 

0.076 

0.446 

5.85 

0.102 

0.578 

5.68 

0.200 

1.148 

5.74 

0.299 

1.840 

6.15 

0.398 

2.596 

6.52 

0.531 

3.830 

7.21 

0.657 

5.120 

7.79 

0.876 

7.970 

9.09 

1.195 

13.610 

11.40 

1.434 

19.518 

13.60 

1.593 

23.870 

14.98 

2.124 

45.000 

21.18 

FERRIC  CHLORIDE. 


m 

A 

A/m 

0.064 

0.387° 

6.05 

0.103 

0.607 

5.89 

0.129 

0.758 

5.87 

0.257 

1.578 

6.14 

0.515 

3.688 

7.16 

1.287 

12.940 

10.06 

1.544 

17.650 

11.40 

2.058 

30.500 

14.82 

2.573 

51.000 

19.82 

CHROMIUM  NITRATE. 


m 

A 

A/m 

0.0467 

0.280° 

6.00 

0.0934 

0.553 

5.90 

0.1868 

1.143 

6.12 

0.3736 

2.493 

6.68 

0.5604 

4.153 

7.41 

0.9340 

8.800 

9.42 

1.1208 

11.570 

10.32 

1.3076 

14.670 

11.22 

1.4944 

19.140 

12.81 

1.8680 

29.500 

15.78 

214  THEORETICAL  CHEMISTRY 

It  will  be  observed  that  in  each  case  there  is  a  minimum  in  the 
molecular  depression  of  the  freezing-point.  The  existence  of  this 
minimum  is  easily  recognized  when  the  molecular  depressions  of 
the  freezing-point  are  plotted  against  the  corresponding  concen- 
trations, as  shown  in  Fig.  66.  The  magnitude  of  the  molecular 
depression  in  the  most  concentrated  solutions  is  very  striking. 
In  the  case  of  a  2.124  molar  solution  of  aluminium  chloride,  the 
molecular  depression  is  21°.18,  while  the  maximum  molecular 
depression  for  a  completely  dissociated  salt  yielding  four  ions  is 
4  X  1°.86  =  7°.44.  The  observed  depression  is  thus  three  times 
the  maximum  theoretical  depression.  This  abnormal  depression 
of  the  freezing-point  may  be  accounted  for  by  assuming  that  the 
dissolved  substance  has  entered  into  combination  with  a  portion 
of  the  water,  thus  removing  it  from  the  role  of  solvent.  The  for- 
mation of  a  loose  molecular  complex  between  one  molecule  of  the 
solute  and  a  large  number  of  molecules  of  water,  acts  as  a  single 
dissolved  unit  in  depressing  the  freezing-point  of  the  pure  sol- 
vent. Evidently  the  total  amount  of  water  present,  which  func- 
tions as  solvent,  is  diminished  by  the  amount  of  water  which  has 
been  appropriated  by  the  solute.  The  abnormalities  observed  in 
the  depression  of  the  freezing-point  of  concentrated  solutions  of 
electrolytes  can  be  explained  by  assuming,  that  the  molecules 
of  solute,  or  the  resulting  ions,  are  in  combination  with  a  number 
of  molecules  of  solvent.  This  hypothesis  is  termed  the  solvate 
theory,  and  the  loose  molecular  complexes  are  called  solvates. 
Since  the  solvate  theory  was  first  proposed,  considerable  evidence 
has  been  accumulated  to  confirm  its  correctness.  Reference  has 
already  been  made  to  the  work  of  Philip  on  the  solubility  of  gases 
in  saline  solutions,  from  which  he  concludes  that  the  dissolved 
salts  enter  into  combination  with  a  portion  of  the  solvent.  The 
experiments  of  Morse  and  the  Earl  of  Berkeley  on  osmotic  pres- 
sure, also  seem  to  point  to  the  solvation  of  the  dissolved  substance. 
If  solvates  exist  in  aqueous  solution,  then  those  solutes  which  ex- 
hibit the  greatest  depressing  action  on  the  freezing-point  of  water, 
should  be  the  ones  which  would  crystallize  from  the  solution  with 
the  greatest  amount  of  water  of  crystallization.  That  there  is 
a  relation  between  freezing-point  depression  and  water  of  crystalli- 


ASSOCIATION,  DISSOCIATION  AND  SOLVATION         215 


Fig.  66. 


216  THEORETICAL  CHEMISTRY 

zation,  is  shown  by  the  curves  in  Fig.  66.  Those  salts  which 
crystallize  without  water  of  crystallization  produce  the  least 
depression  of  the  freezing-point.  Those  crystallizing  with  two 
molecules  of  water  are  next  in  order,  and  then  follow  those  crystal- 
lizing with  four  and  six  molecules  respectively.  Similar  relations 
are  found  to  hold  for  the  corresponding  bromides,  iodides  and  ni- 
trates. We  conclude,  therefore,  that  those  solutes  which  crystallize 
with  the  same  amount  of  water  produce  approximately  the  same 
depression  of  the  freezing-point  of  water.  Another  line  of  evidence 
bearing  upon  the  solvate  theory,  is  that  furnished  by  the  molec- 
ular elevation  of  the  boiling-point  of  concentrated  solutions  of 
electrolytes.  When  boiling-point  elevations  are  plotted  against 
corresponding  concentrations,  the  curves  present  well-defined 
minima,  but  at  concentrations  greater  than  those  corresponding 
to  the  minima  in  the  freezing-point  curves.  This  is  in  agreement 
with  predictions  based  upon  the  solvate  theory,  since  at  higher 
temperatures,  the  solvates  would  naturally  be  less  stable  and, 
therefore,  a  greater  concentration  would  be  required  to  produce  a 
sufficient  amount  of  these  substances  to  cause  a  change  in  the 
direction  of  the  curve.  Still  further  evidence  in  favor  of  the 
solvate  theory  has  been  furnished  by  Jones  and  his  co-workers  * 
from  a  study  of  the  absorption  spectra  of  solutions  of  colored 
salts.  All  of  these  investigations  point  to  the  existence  of  solvates 
in  solutions,  the  phenomenon  being  by  no  means  confined  to 
aqueous  solutions  alone. 

• 

PROBLEMS. 

1.  At  18°  C.  a  0.5  molar  solution  of  NaCl  is  74.3  per  cent  dissociated. 
What  would  be  the  osmotic  pressure  of  the  solution  in  atmospheres  at 
18°  C.?  Ans.  20.79. 

2.  A  solution  containing  3  mols  of  cane  sugar  per  liter  was  found  by 
the  plasmolytic  method  to  be  isotonic  with  a  solution  of  potassium  nitrate 
containing  1.8  mols  per  liter.    What  is  the  degree  of  ionization  of  the 
potassium  nitrate?  Ans.   67  per  cent. 

*  Monographs,  Nos.  60,  130,  and  160,  Carnegie  Institution  of  Washing- 
ton; Am.  Chem.  Jour.,  49,  1  (1913). 


ASSOCIATION,  DISSOCIATION  AND  SOLVATION         217 

3.  A  solution  containing  1.9  mols  of  calcium  chloride  per  liter  is  isotonic 
with  a  solution  of  glucose  containing  4.05  mols  per  liter.     What  is  the 
degree  of  ionization  of  the  calcium  chloride?  Ans.   56.6  per  cent. 

4.  At  0°  C.  the  vapor  pressure  of  water  is  4.620  mm.  and  of  a  solution     \/ 
of  8.49  grams  of  NaN03  in  100  grams  of  water  4.483  mm.     Calculate  the 
degree  of  ionization  of  NaN03.     •  Ans.   64.9  per  cent. 

5.  At  0°  C.  the  vapor  pressure  of  water  is  4.620  mm.  and  that  of  a   yN 
solution  of  2.21  grams  of  CaCl2  in  100  grams  of  water  is  4.583  mm.     Cal- 
culate the  apparent  molecular  weight  and  the  degree  of  ionization  of 

CaCl2.  Ans.   M  =  52.5,     a  =  55.7  per  cent. 

' 


6.  The  boiling-point  of  a  solution  of  0.4388  gram  of  sodium  chloride       r 
in  100  grams  of  water  is  100°.074  C.     Calculate  the  apparent  molecular 
weight  of  the  sodium  chloride  and  its  degree  of  ionization.    K  =  5.2. 

Ans.   M  =  30.84,     a  =  89.7  per  cent. 

7.  The  boiling-point  of  a  solution  of  3.40  grams  of  BaCl2  in  100  grams    y} 
of  water  is  100°.208  C.    K  =  5.2.    What  is  the  degree  of  ionization  of 
the  BaCl2?  Ans.   72.5  per  cent. 

8.  At  100°  C.  the  vapor  pressure  of  a  solution  of  6.48  grams  of  ammo- 


nium  chloride  in  100  grams  of  water  is  731.4  mm.    K  =  5.2.    What  is  the 
boiling-point  of  the  solution?  Ans.   101°.086  C. 

9.  A  solution  of  1  gram  of  silver  nitrate  in  50  grams  of  water  freezes 
at  —  0°.348  C.     Calculate  to  what  extent  the  salt  is  ionized  in  solution. 
K  =  18.6.  Ans.   59  per  cent. 

10.  A  solution  of  NaCl  containing  3.668  grams  per  1000  grams  of 
water  freezes  at  —  0°.2207  C.     Calculate  the  degree  of  ionization  of  the 
salt.    K  =  18.6.  Ans.   89.2  per  cent. 

11.  The  freezing-point  of  a  solution  of  barium  hydroxide  containing    / 
1  mol  in  64  liters  is  —  0°.0833  C.    What  is  the  concentration  of  hydroxyl  ' 
ions  in  the  solution?    Take  K  =  18.9  for  concentrations  in  mols  per 
liter.  Ans.   0.0284  gm.-ion  per  liter. 

12.  The  vapor  pressure  of  water  at  0°  C.  is  4.620  mm.,  and  the  lower-  ^ 
ing  of  the  vapor  pressure  produced  by  dissolving  5.64  grams  of  sodium 
chloride  in  100  grams  of  water  is  0.142  mm.    What  is  the  freezing-point 

of  the  solution?    K  =  18.6.  Ans.    -  3°.177  C. 

13.  A  solution  containing  0.834  gram  Na2S04  per  1000  grams  of  water  Y 
freezes  at  —  0°.280  C.    Assuming  dissociation  into  3  ions,  calculate  the 


218  THEORETICAL  CHEMISTRY 

degree  of  ionization  and  the  concentrations  of  the  Na"  and  S04"  ions. 
K  =  18.6. 

Ans.   a  =  78.2  per  cent;   cone.  Na*  =  0.0918  gm.-ion  per  liter;  cone. 
S04"  =  0.0459  gm.-ion  per  liter. 

14.  A  solution  of  copper  chloride  containing  3  mols  of  salt  per  liter 
depresses  the  freezing-point  of  water  25°. 500.  Taking  K  =  18.9  for  con- 
centrations in  mols  per  liter,  calculate  the  apparent  molecular  weight  of 
copper  chloride.  How  do  you  account  for  the  result? 


CHAPTER  X. 
COLLOIDAL  SOLUTIONS. 

Crystalloids  and  Colloids.  In  the  course  of  his  investigations 
on  diffusion  in  solutions,  Thomas  Graham  *  drew  a  distinction 
between  two  classes  of  solutes,  which  he  termed  crystalloids  and 
colloids  respectively.  Crystalloids,  as  the  name  implies,  can  be 
obtained  in  the  crystalline  form:  to  this  class  belong  nearly  all 
of  the  acids,  bases  and  salts.  Colloids,  on  the  other  hand,  are 
generally  amorphous,  such  substances  as  albumen,  starch  and 
caramel  being  typical  of  the  class.  Because  of  the  gelatinous 
character  of  many  of  the  substances  in  this  class,  Graham  termed 
them  colloids,  OcoAXa  =  glue,  and  etSos  =  form).  The  differ- 
ences between  the  two  classes  are  most  apparent  in  the  physical 
properties  of  their  solutions.  Thus,  crystalloids  diffuse  much 
more  rapidly  than  colloids;  the  velocity  of  diffusion  of  caramel 
being  nearly  100  times  slower  than  that  of  hydrochloric  acid  at 
10°  C.  While  crystalloids  exert  osmotic  pressure,  lower  the  vapor 
pressure,  and  depress  the  freezing-point  of  the  solvent,  colloids 
have  very  little  effect  upon  the  properties  of  the  solvent.  The 
marked  differences  in  the  rates  of  diffusion  of  crystalloids  and 
colloids  render  their  separation  comparatively  easy.  If  a  solution 
containing  both  crystalloids  and  colloids  be  placed  in  a  vessel 
over  the  bottom  of  which  is  stretched  a  colloidal  membrane,  such 
as  parchment,  and  the  whole  is  immersed  in  pure  water,  the 
crystalloids  will  pass  through  the  membrane,  while  the  colloids 
will  be  left  behind.  This  process  was  termed  by  Graham,  dialysis 
(StaXuo-ts  =  a  separation),  and  the  apparatus  employed  to  effect 
such  a  separation  was  called  a  dialyzer.  When  a  solution  of 
sodium  silicate  is  added  to  an  excess  of  hydrochloric  acid,  the 
products  of  the  reaction,  silicic  acid  and  sodium  chloride,  remain 
in  solution.  When  the  mixture  is  placed  in  a  dialyzer,  the  sodium 
*  Lieb.  Ann.,  121,  1  (1862). 
219 


220 


THEORETICAL  CHEMISTRY 


chloride  and  the  hydrochloric  acid,  being  crystalloids,  diffuse 
through  the  membrane  of  the  dialyzer,  leaving  behind  the  colloidal 
silicic  acid.  The  terms  crystalloid  and  colloid,  as  used  at  the 
present  time,  have  acquired  a  different  meaning  from  that  assigned 
to  them  by  Graham.  t 

The  terms  are  now  considered  to  refer,  not  so  much  to  different 
classes  of  substances,  as  to  different  states  which  almost  all  sub- 
stances can  assume  under  certain  conditions. 

Colloidal  Solutions.  A  colloidal  solution  is  one  in  which  the 
solute  is  a  colloid,  although  the  latter  may  not  be  included  among 
the  substances  classified  as  such  by  Graham.  For  example, 
arsenious  sulphide,  ferric  hydroxide  or  finely-divided  gold  may 
form  colloidal  solutions.  In  bringing  such  substances  into  the 
colloidal  state,  mere  agitation  with  water  will  not  suffice,  but  some 
indirect  method  must  be  employed.  Therefore,  the  precipita- 
tion of  these  substances  from  their  colloidal  solutions  cannot  be 
reversed,  and  consequently  they  are  known  as  irreversible  colloids, 
in  distinction  to  such  substances  as  albumen,  gum,  etc.,  which 
are  soluble  in  water,  and  hence  are  termed  reversible  colloids. 

Density  of  Colloidal  Solutions.  The  density  of  colloidal 
solutions  cannot  be  calculated  from  the  densities  of  the  solute 
and  solvent,  the  observed  value  being,  in  general,  greater  than 
the  calculated  value.  For  example,  when  gelatine  is  dissolved 
in  water  it  contracts  in  volume,  as  is  shown  by  the  following 
figures,  giving  the  volume  of  gelatine  solution  which  can  be 
obtained  from  one  cubic  centimeter  of  water. 


Percentage 
Concentration. 

Volume  of 
Solution. 

10 
25 
50 

0.96069 
0.93748 
0.90201 

Osmotic  Pressure  of  Colloidal  Solutions.  The  osmotic  pres- 
sure of  colloidal  solutions  is  very  small.  This  is  what  we  should 
expect  with  solutions  of  substances  which  exhibit  a  slow  rate  of 
diffusion.  As  has  been  pointed  out,  diffusion  is  closely  connected 


COLLOIDAL  SOLUTIONS 


221 


with  osmotic  pressure;  hence,  if  the  rate  of  diffusion  is  slow,  the 
osmotic  pressure  exerted  by  the  solution  should  be  small.  In 
some  cases  the  osmotic  pressure  is  so  small  as  to  escape  detection. 
The  experimental  determination  of  the  osmotic  pressure  of  col- 
loidal solutions  is  complicated  by  the  difficulty  of  removing  the 
last  traces  of  electrolytes  from  the  colloid.  Owing  to  their  great 
osmotic  activity,  the  presence  of  the  merest  trace  of  electrolytes 
may  mask  the  osmotic  effect  of  the  colloid.  In  the  recent  in- 
vestigations by  Duclaux  *  great  care  has  been  taken  to  purify 
the  colloids,  and  the  values  given  by  him  may  be  considered  to 
represent  very  closely  the  actual  osmotic  pressures  of  the  colloidal 
solutions.  The  following  table  gives  the  results  obtained  by 
Duclaux  with  colloidal  solutions  of  ferric  hydroxide. 


Cone.  Fe(OH)3 
Per  cent. 

Pressure  in  cm. 
of  Water. 

1.08 

0.8 

2.04 

2.8 

3.05 

5.6 

5.35 

12.5 

8.86 

22.6 

Inspection  of  the  table  shows  that,  even  in  the  most  concen- 
trated solution,  the  osmotic  pressure  is  very  small.  Furthermore, 
it  is  apparent  that  although  the  osmotic  pressure  increases  with 
the  concentration,  the  variables  are  not  proportional.  Observa- 
tions on  the  variation  of  the  osmotic  pressure  of  colloidal  solu- 
tions with  temperature,  show  that,  in  general,  as  the  temperature 
is  raised  the  pressure  increases  at  a  more  rapid  rate  than  that 
required  by  the  law  of  Gay-Lussac.  It  has  also  been  observed, 
that  the  value  of  the  osmotic  pressure  of  gelatine  solutions  at 
ordinary  temperatures  can  be  increased  by  maintaining  the  solu- 
tions at  a  higher  temperature  for  a  short  time  and  then  cooling 
to  the  initial  temperature.  After  standing  for  several  days  at 
.the  original  temperature,  however,  the  osmotic  pressure  of  the 
solution  returns  to  its  former  value.  This  phenomenon  would 

*  Compt.  rend.,  140,  1544  (1905);  Jour.  Chem.  Phys.,  7,  405  (1909). 


222  THEORETICAL  CHEMISTRY 

seem  to  indicate  that  the  osmotic  pressure  of  colloidal  solutions 
is  not  completely  denned  by  the  two  variables,  temperature  and 
concentration.  It  has  been  suggested  that  the  degree  of  aggre- 
gation of  the  colloid  is  partially  dependent  upon  the  temperature; 
the  molecular  aggregates  tending  to  break  up  as  the  temperature 
is  raised,  thus  increasing  the  number  of  dissolved  units  and 
therefore  causing  a  corresponding  increase  in  the  osmotic  pres- 
sure. 

Molecular  Weight  of  Colloids.  We  have  already  learned  that 
the  knowledge  of  the  osmotic  pressure  of  a  solution  enables  us  to 
calculate  the  molecular  weight  of  the  solute,  provided  the  solu- 
tion is  dilute  and  obeys  the  gas  laws.  As  we  have  seen,  other 
factors  than  concentration  and  temperature  determine  the  osmotic 
pressure  of  colloidal  solutions,  so  that  we  are  not  justified  in 
attempting  to  calculate  the  molecular  weight  of  the  dissolved 
substance  from  the  observed  value  of  the  osmotic  pressure  of  its 
solutions.  Values  for  the  molecular  weight  of  colloids  calculated 
from  their  effect  on  the  vapor  pressure,  the  boiling-point,  and  the 
freezing-point  of  the  solvent  are  also  untrustworthy,  since  the 
same  factors  which  influence  the  osmotic  pressure  undoubtedly 
affect  these  related  properties.  The  small  values  of  the  osmotic 
pressure  of  colloidal  solutions  has  been  interpreted  as  an  indication 
of  the  large  molecular  weight  of  the  dissolved  substance. 

Electrical  and  Magnetic  Properties  of  Colloidal  Solutions.  It 
was  first  observed  by  Linder  and  Picton  *  that  when  the  terminals 
of  an  electric  battery  are  connected  to  two  platinum  electrodes 
dipping  into  a  colloidal  solution  of  arsenious  sulphide,  there  is  a 
gradual  migration  of  the  colloid  to  the  positive  pole.  A  similar 
experiment  with  a  solution  of  colloidal  ferric  hydroxide  resulted 
in  the  transport  of  the  dissolved  colloid  to  the  negative  pole.  It 
follows,  therefore,  that  the  particles  of  colloidal  arsenious  sulphide 
are  negatively  charged,  while  those  of  colloidal  ferric  hydroxide 
carry  a  positive  charge.  It  has  been  found  that  all  colloids  carry 
an  electric  charge.  In  the  following  table,  some  typical  colloids 
are  classified  according  to  the  character  of  their  electrification  in 
aqueous  solution. 

*  Jour.  Chem.  Soc.,  61,  148  (1892). 


COLLOIDAL  SOLUTIONS 


223 


E  lectro-positi  ve. 

Electro-negative. 

Metallic  hydroxides 
Methyl  violet 
Methylene  blue 
Magdala  red 
Bismark  brown 
Haemoglobin 

All  the  metals 
Metallic  sulphides 
Aniline  blue 
Indigo 
Eosine 
Starch 

The  nature  of  the  charge  varies  with  the  solvent  used,  colloids 
in  solution  in  turpentine,  for  example,  having  charges  opposite 
to  those  in  water. 

W.  B.  Hardy  *  has  found  that  the  direction  of  migration  of 
albumen,  modified  by  heating  to  100°  C.,  is  dependent  upon  the 
reaction  of  the  medium  in  which  it  is  dissolved.  A  very  small 
quantity  of  free  base  causes  the  particles  of  albumen  to  move 
toward  the  positive  electrode,  while  the  addition  of  an  equally 
small  amount  of  acid  results  in  a  reversal  of  the  direction  of 
migration.  Similar  reversals  of  charge  have  been  observed  by 
Burton  t  in  colloidal  solutions  of  gold  and  silver. 

When  small  amounts  of  aluminium  sulphate  are  added  to 
colloidal  solutions  of  these  metals,  the  charge  is  gradually  neu- 
tralized and  eventually  the  colloidal  particles  acquire  a  reversed 
charge. 

The  electrical  conductance  of  colloidal  solutions  is  considerably 
smaller  than  that  of  solutions  of  electrolytes.  Colloidal  solutions 
of  substances  which  are  magnetic  in  the  free  state,  such  as  iron 
and  nickel,  acquire  magnetic  properties. 

Colloidal  Suspensions.  When  a  difference  of  potential  is  es- 
tablished between  two  electrodes  immersed  in  a  suspension  of 
finely-divided  quartz  or  shellac,  the  suspended  particles  move 
toward  the  positive  electrode.  This  phenomenon  is  called  cola- 
phoresis.  The  behavior  of  suspensions  when  placed  in  an  electric 
field  is  thus  seen  to  resemble  that  of  colloidal  solutions  under  similar 
conditions.  In  fact,  there  is  abundant  evidence  to  strengthen 
the  view  that  colloidal  solutions  and  simple  suspensions  are 

*  Jour.  Physiol.,  24,  288  (1899). 
t  Phil.  Mag.,  12,  472  (1906). 


224 


THEORETICAL  CHEMISTRY 


closely  related.  Suspensions  of  all  grades  exist,  from  those  in 
which  the  suspended  particles  are  coarse-grained  and  visible  to 
the  naked  eye  down  to  those  in  which  a  high-power  microscope 
is  required  to  render  the  suspended  particles  visible.  Colloidal 
solutions  have  also  been  shown  to  be  non-homogeneous,  the 
presence  of  discrete  particles  being  revealed  by  means  of  the 
ultramicroscope.  It  follows,  therefore,  that  the  size  of  the  particles 
in  solution  determines  whether  a  substance  is  to  be  considered  as 
a  colloid  or  not.  At  one  extreme  we  have  true  solutions  in  which 
no  lack  of  homogeneity  can  be  detected,  even  by  the  ultramicro- 
scope, and  at  the  other  extreme  we  have  coarse-grained  suspen- 
sions, in  which  the  particles  are  visible  to  the  naked  eye.  Between 
these  two  limits  all  possible  degrees  of  subdivision  are  possible 
and  it  is  a  very  difficult  matter  to  draw  sharp  lines  of  distinction 
between  true  solutions  and  colloidal  solutions  on  the  one  hand, 
and  between  colloidal  solutions  and  suspensions  on  the  other. 
One  of  the  most  satisfactory  schemes  of  classification  is  that  of 
von  Weimarn  and  Wo.  Ostwald.*  Because  of  the  fact  that  sus- 
pensions, colloidal  solutions,  and  true  solutions  represent  varying 
degrees  of  dispersion  of  the  solute,  all  three  types  of  system  are 
termed  by  these  authors,  dispersoids.  The  dispersoids  are  classi- 
fied as  shown  in  the  accompanying  diagram. 

DISPERSOIDS. 


Coarse  disp 
pensions, 
e 
Magnitude 
greater  t 

•ersions  (sus- 
emulsions, 
tc.). 
of  particles 
hanO.1  /i.* 

Colloidal  solutions. 
Magnitude  of  particles 
between  0.1  /i  and  1  MM- 

Decreasing  degree  of 
"  colloidity." 

Increasing  degree  of  dis- 
persion. 

*  l/i=  1  micron  =  0.001  mm. 

'  Koll.  Zeitschrift,  3,  26  (1908). 


Molecular         Ionic 
dispersoids.  dispersoids 


Magnitude  of  particles, 
about  IMJU  and  smaller. 


COLLOIDAL  SOLUTIONS  225 

The  Ultramicroscope.  When  a  narrow  beam  of  sunlight  is 
admitted  into  a  darkened  room,  the  dust  particles  in  its  path  are 
rendered  visible  by  the  scattering  of  the  light  at  the  surface  of  the 
particles.  If  the  air  of  the  room  is  free  from  dust,  no  shining 
particles  will  be  seen  and  the  space  is  said  to  be  "optically  void." 
When  the  particles  of  dust  are  very  minute,  the  beam  of  light 
acquires  a  bluish  tint.  The  blue  color  of  the  sky  is  thus  attrib- 
uted to  the  presence  of  extremely  fine  particles  of  dust  in  the  air 
together  with  minute  drops  of  condensed  gases  in  the  upper  regions 
of  the  atmosphere. 

The  visibility  of  a  beam  of  light  due  to  the  scattering  effect  of 
minute  particles,  is  known  as  the  "  Tyndall  phenomenon. "  Almost 
all  colloidal  solutions  exhibit  this  phenomenon  when  a  powerful 
beam  of  light  is  passed  through  them,  thus  proving  the  presence 
of  discrete  particles  in  the  solutions. 

The  ultramicroscope  is  an  instrument  devised  by  Siedentopf 
and  Zsigmondy  *  for  the  detection  of  colloidal  particles  much  too 
small  to  be  seen  by  the  naked  eye.  A  powerful  beam  of  light 
issuing  from  a  horizontal  slit  is  brought  to  a  focus  within  the 
colloidal  solution  under  examination  by  means  of  a  microscope 
objective,  and  this  image  is  viewed  through  a  second  micro- 
scope, the  axis  of  which  is  at  right  angles  to  the  path  of  the 
beam. 

When  examined  in  this  way  a  colloidal  solution  appears  to  be 
swarming  with  brilliant  particles,  moving  rapidly  in  a  dark  field; 
whereas  a  true  solution,  if  properly  prepared,  appears  optically 
void.  With  the  ultramicroscope  it  is  possible  to  count  the  num- 
ber of  particles  present  in  a  given  volume  of  a  colloidal  solution, 
and  by  means  of  a  chemical  analysis,  the  mass  of  colloid  in  this 
volume  can  be  determined.  Hence  the  average  mass  of  each 
particle  can  be  calculated.  If  the  particles  be  assumed  to  be 
spherical  in  shape  and  to  have  the  same  density  as  larger  masses 
of  the  same  substance,  we  can  calculate  the  volume  of  a  single 
particle  and  from  this  its  diameter.  Thus,  Burton  f  in  his  experi- 
ments on  colloidal  solutions  of  gold,  silver  and  platinum,  found 

*  Colloids  and  the  Ultramicroscope,  R.  Zsigmondy. 
t  Phil.  Mag.,  ii,  425  (1906). 


226  THEORETICAL  CHEMISTRY 

the  average  diameter  of  the  colloidal  particles  to  range  from  0.2 
to  0.6  micron. 

The  smallest  particles  which  can  be  detected  with  the  ultramicro- 
scope  are  those  of  red  colloidal  gold  solutions,  the  average  cal- 
culated diameter  of  a  single  particle  being  6  millimicrons.  It  is 
interesting  to  note  that  the  magnitude  of  these  smallest  particles 
revealed  by  the  ultramicroscope  is  nearly  ten  times  the  average 
magnitude  of  the  molecule  of  chloroform. 

The  Brownian  Movement.  When  a  colloidal  solution  or 
suspension,  containing  particles  about  1  micron  in  diameter,  is 
examined  under  the  microscope,  the  particles  will  be  seen  to  be 
in  rapid  oscillatory  motion.  This  interesting  phenomenon  was 
first  observed  with  grains  of  pollen  by  the  English  botanist, 
Robert  Brown.  As  the  size  of  the  particles  is  diminished  the 
amplitude  of  the  oscillations  increases,  until,  when  diameters  of 
about  20  millimicrons  are  reached,  the  particles  move  over  a 
rectilinear  path  of  nearly  20  microns  before  changing  their  direc- 
tion. The  particles  move  with  high  velocities  and  the  oscillation 
is  perpetual.  The  recent  experimental  researches  of  Perrin  *  on 
the  Brownian  movement  have  shown  that  the  behavior  of  these 
particles  is  in  exact  agreement  with  the  requirements  of  the 
kinetic  theory  for  molecules  of  the  same  dimensions  as  the  particles. 
The  complete  analogy  between  visible  colloidal  particles  and 
molecules  has  recently  been  demonstrated  by  Perrin  f  in  a  series 
of  experiments  of  unusual  beauty.  The  Brownian  movement 
may  be  looked  upon  as  a  visible  manifestation  of  the  ceaseless 
vibratory  motion  of  the  molecules  of  matter  assumed  in  the  de- 
velopment of  the  kinetic  theory. 

Suspensoids  and  Emulsoids.  When  colloidal  solutions  are 
considered  with  reference  to  precipitating  or  coagulating  agents, 
they  may  be  divided  conveniently  into  two  classes,  as  follows :  — 
(1)  Colloids  which  resemble  suspensions,  or  suspensaids,  and  (2) 
colloids  which  resemble  emulsions,  or  emulsoids.  Colloids  belong- 
ing to  the  suspensoid  class  give  with  water,  non-viscous,  non- 
gelatinizing  mixtures,  which  are  readily  coagulated  on  the 

*  "The  Brownian  Movement  and  Molecular  Reality,"  by  J.  Perrin. 
t  Chem.  News,  106,  189,  203,  215  (1912). 


COLLOIDAL  SOLUTIONS  227 

addition  of  small  amounts  of  electrolytes.  Colloids  belonging  to 
the  emulsoid  class  give  with  water,  viscous,  gelatinizing  mixtures, 
which  are  not  readily  coagulated  by  electrolytes.  The  suspen- 
soids  are  irreversible  colloids  while  the  emulsoids  are  reversible 
colloids. 

Coagulation  of  Suspensoids.  Colloidal  solutions  of  arsenious 
sulphide  and  ferric  hydroxide  are  easily  precipitated  by  the 
addition  of  a  very  small  amount  of  an  electrolyte.  In  like 
manner,  the  sedimentation  of  suspensions,  such  as  kaolin  in  water, 
is  promoted  by  the  addition  of  electrolytes.  On  the  other  hand, 
non-electrolytes  have  no  effect  upon  the  stability  of  a  solution  of 
a  suspensoid  or  a  simple  suspension.  The  phenomenon  of  the 
coagulation  of  suspensoids  has  been  carefully  investigated  by 
Freundlich.*  He  has  found  that  an  amount  of  electrolyte  which 
is  incapable  of  bringing  about  an  instantaneous  coagulation,  may 
become  effective  after  an  interval  of  time.  He  has  also  shown 
that  the  total  quantity  of  electrolyte  required  to  coagulate  the 
suspensoid  completely,  depends  upon  whether  the  electrolyte  is 
added  all  at  one  time  or  in  successive  portions.  In  order  to  com- 
pare the  coagulating  action  of  various  electrolytes,  Freundlich  pro- 
posed the  following  procedure,  which  prevents  the  possibility  of 
irregularities  due  to  the  time  factor:  —  To  20  cc.  of  the  solution 
of  the  suspensoid,  2  cc.  of  the  solution  of  the  electrolyte  are  added, 
the  solution  being  shaken  vigorously;  the  mixture  is  then  set 
aside  for  two  hours,  after  which  a  small  portion  is  filtered  off,  and 
the  filtrate  is  examined  for  the  suspensoid.  In  the  following  table 
some  of  the  results  obtained  by  Freundlich  with  colloidal  solu- 
tions of  arsenious  sulphide  are  given.  The  data  represent  the 
minimum  concentration  for  each  electrolyte  which  produced 
coagulation  in  two  hours. 

It  will  be  seen  that  very  small  amounts  of  the  electrolytes  are 
required  to  coagulate  the  suspensoid,  and  further,  that  the  coagu- 
lating power  of  an  electrolyte  is  dependent  upon  the  charge  of  the 
positive  ion.  The  greater  the  charge,  the  smaller  is  the  quantity 
of  electrolyte  required  to  produce  coagulation.  Similar  results 
were  obtained  by  Freundlich  with  colloidal  solutions  of  ferric 
*  Zeit.  phys.  Chem.,  44,  131  (1903). 


228 


THEORETICAL  CHEMISTRY 


Electrolyte. 

Concentration 
in  Millimols 
per  Liter. 

NaCL. 

71  2 

KNO3.. 

69  8 

1  K2SO4  .  . 

91.5 

NH4C1... 

59.1 

HC1 

42  9 

MgCl2. 

1  00 

MgSO4  

1  13 

Ca(NO3)2..    . 

0  95 

BaCl2  

0.96 

ZnSO4  

1.13 

A1C13 

0  13 

A1(NO3)3 

0  14 

|Ce2(SO4)3.     . 

0  13 

hydroxide,  the  coagulating  power  of  an  electrolyte  being  largely 
determined  by  the  charge  of  the  negative  ion.  The  significance 
of  the  relation  between  ionic  charge  and  coagulating  power  was 
first  pointed  out  by  Hardy,*  who  formulated  the  following  rule:  — 
The  coagulation  of  a  colloidal  solution  is  determined  by  that  ion  of 
an  added  electrolyte  which  has  an  electric  charge  opposite  in  sign  to 
that  of  the  colloidal  particles.  It  has  already  been  pointed  out 
that  colloidal  particles  of  arsenious  sulphide  are  negatively  charged; 
hence,  according  to  Hardy's  rule,  the  positive  ions  of  the  electrolyte 
added  will  condition  the  coagulation  of  the  suspensoid.  The 
experiments  of  Freundlich  confirm  this  prediction.  Again  the 
validity  of  the  rule  is  illustrated  by  Hardy's  experiments  on  egg 
albumen  to  which  reference  has  been  made.  In  a  slightly  alkaline 
medium,  albumen  i£  electro-negative,  while  in  a  slightly  acid 
medium  the  sign  of  the  electric  charge  is  reversed.  It  is  found, 
therefore,  that  in  a  faintly  alkaline  medium,  A12(S04)3  is  more 
effective  than  Na^SC^  in  producing  coagulation  of  the  albumen, 
while  the  coagulating  power  of  MgS04  is  intermediate  between 
the  other  two  sulphates.  On  the  other  hand,  in  a  slightly  acid 
medium,  the  albumen  is  coagulated  equally  well  by  all  three 
*  Zeit.  phys.  Chem.,  33,  385  (1900). 


COLLOIDAL  SOLUTIONS  229 

salts.  Similarly,  while  BaCl2  is  more  efficient  than  Na^SC^  in 
coagulating  the  albumen  in  an  alkaline  medium,  the  order  of 
efficiency  is  reversed  in  an  acid  medium.  These  results,  taken 
together  with  the  fact  that  non-electrolytes  are  without  effect, 
show  that  the  coagulation  of  suspensoids  is  essentially  an  ionic 
process.  This  view  is  strengthened  by  the  fact,  that  when  various 
electrolytes  having  a  common  positive  ion  are  used  as  coagulants, 
their  efficiency  varies  directly  with  their  degree  of  ionization. 
Since  the  coagulation  of  a  suspensoid  results  from  a  neutraliza- 
tion of  the  electric  charge  of  the  colloid  by  the  opposite  charge  of 
an  ion  of  an  added  electrolyte,  it  follows  that  the  precipitated 
colloid  or  hydrogel,  as  it  is  called,  should  contain  either  the  acid 
or  basic  portion  of  the  electrolyte.  The  experiments  of  Linder 
and  Picton  *  have  shown  this  to  be  the  case.  Thus,  when  a 
colloidal  solution  of  arsenious  sulphide  is  coagulated  by  BaCl2, 
the  hydrogel  is  found  to  contain  barium.  Continued  washing 
with  water  fails  to  remove  the  barium,  but  by  treating  the  hydro- 
gel  with  a  solution  of  another  electrolyte  it  may  be  replaced 
metathetically  by  another  basic  element.  The  retention  of  the 
basic  portion  of  the  precipitant  by  the  hydrogel  is  considered  to 
be  a  case  of  adsorption. 

Reciprocal  Coagulation.  A  further  consequence  of  the  elec- 
trical theory  of  coagulation  is,  that  when  two  oppositely-charged 
colloids  are  mixed,  they  should  precipitate  each  other,  and  the 
resulting  hydrogel  should  contain  both  colloids.  Experiments 
carried  out  by  Biltz  f  have  confirmed  these  predictions.  He 
showed  that  when  a  solution  of  a  positively-charged  colloid  is 
added  to  a  solution  of  a  negatively-charged  colloid,  precipitation 
occurs,  unless  the  quantity  of  the  added  colloid  is  either  relatively 
very  large  or  very  small.  He  also  showed  that  when  two  colloids 
of  the  same  electrical  sign  are  mixed  no  coagulation  results. 

Precipitation  of  Emulsoids.  Emulsoids  are  by  no  means  as 
sensitive  to  the  presence  of  electrolytes  as  are  suspensoids. 
Further,  the  precipitation  of  an  emulsoid  by  a  neutral  alkali 
salt  is  reversible  while  the  corresponding  precipitation  of  a  sus- 

*  Jour.  Chem.  Soc.,  67,  63  (1895). 
f  Berichte,  37,  1095  (1904). 


230  THEORETICAL  CHEMISTRY 

pensoid  is  irreversible.  As  has  been  seen,  the  precipitation  of  a 
suspensoid  by  an  electrolyte  is  essentially  an  electrical  phenomenon, 
whereas  the  precipitation  of  an  emulsoid  by  a  neutral  alkali  salt 
does  not  involve  the  neutralization  of  electric  charges,  but  is  a 
phenomenon  similar  to  "salting  out,"  familiar  to  the  organic 
chemist. 

Protective  Colloids.  The  precipitating  action  of  electrolytes 
on  suspensoids  may  be  inhibited  by  adding  to  the  solution  of  the 
suspensoid  a  reversible  colloid.  The  protective  action  of  a  re- 
versible colloid  is  not  due,  as  might  be  supposed,  to  the  increased 
viscosity  of  the  medium  and  the  resultant  resistance  to  sedimenta- 
tion, since  amounts  of  a  reversible  colloid,  too  minute  to  produce 
any  appreciable  increase  in  the  viscosity  of  the  medium,  can  pre- 
vent coagulation.  Thus,  Bechhold  *  has  shown  that  while  a 
mixture  of  1  cc.  of  a  suspension  of  mastic  and  1  cc.  of  a  0.1  molar 
solution  of  MgS04  diluted  to  3  cc.  with  water,  is  completely 
coagulated  in  15  minutes,  no  coagulation  occurs  within  24  hours, 
if  two  drops  of  a  1  per  cent  solution  of  gelatine  be  added  before 
diluting  to  3  cc. 

Gum  arabic  and  ox-blood  serum  exert  a  similar  protective  action 
when  added  to  a  suspension  of  mastic.  The  protective  power  of 
reversible  colloids  differs  widely  and  Zsigmondy  f  has  attempted 
to  make  this  the  basis  of  a  method  of  classification  of  the  proteins. 
A  red  solution  of  colloidal  gold  becomes  blue  on  the  addition  of  a 
small  amount  of  sodium  chloride,  owing  to  the  increase  in  the  size 
of  the  colloidal  particles.  Various  proteins  when  added  to  the 
red  solution  of  colloidal  gold  protect  the  colloidal  particles 
from  coagulation  by  the  sodium  chloride,  no  change  in  color 
following  the  addition  of  the  electrolyte.  A  definite  amount  of 
each  protein  is  required  to  prevent  the  change  from  red  to  blue  in 
the  color  of  the  colloidal  solution.  In  employing  this  color  change 
as  a  means  of  differentiating  protein  substances,  Zsigmondy  intro- 
duced the  "gold  number,"  which  may  be  defined  as  the  weight  in 
milligrams  of  the  protein  which  is  just  insufficient  to  prevent  the 
change  from  red  to  blue  in  10  cc.  of  a  solution  of  colloidal  gold 

*  Zeit.  phys.  Chem.,  48,  408  (1904). 
t  Zeit.  analyt.  Chem.,  40,  697  (1901). 


COLLOIDAL  SOLUTIONS 


231 


after  the  addition  of  1  cc.  of  a  10  per  cent  solution  of  sodium 
chloride.  The  following  table  gives  the  gold  numbers  of  a  few 
proteins. 


Substance. 

Gold  Number. 

Gelatine 

0  005  to  0  01 

Globulin  

0.02    to  0.05 

Egg  albumen 

0  15    to  0  25 

Wheat  starch  . 

About  4.0      to  6  0 

Potato  starch  

About  25  .  0 

Surface  Energy  of  Colloids.  In  almost  all  colloidal  solutions 
there  exists  a  difference  of  potential  between  the  particles  of  col- 
loid and  the  surrounding  medium.  The  importance  of  this  factor 
in  interpreting  the  behavior  of  colloids  has  already  been  empha- 
sized. Another  factor  of  equal  importance  in  connection  with 
colloidal  phenomena  is  that  which  depends  upon  the  enormous 
surface  of  contact  between  the  colloid  and  the  surrounding 
medium.  There  is  an  abundance  of  evidence  showing  that  a 
colloidal  solution  is  non-homogeneous,  or  in  other  words,  that  it 
is  essentially  a  suspension  of  finely-divided  particles  in  a  fluid 
medium.  The  division  and  subdivision  of  matter  results  in  an 
immense  increase  of  its  surface  area,  and  to  bring  about  such 
comminution  requires  an  expenditure  of  energy.  In  a  colloidal 
solution  this  energy  is  stored  up  hi  the  colloidal  particles  in  the 
form  of  surface  energy,  which  may  be  defined  as  the  product  of 
surface  area  and  surface  tension. 

For  example,  suppose  1  cc.  of  a  substance  to  be  reduced  to 
cubical  particles  measuring  0.1  M  on  each  edge,  and  let  the  particles 
be  suspended  hi  water  at  17°  C.  The  total  energy  involved  can 
be  calculated  as  follows:  —  The  volume  of  a  single  particle  is 
0.1  M3  or  1  X  10~15  cc.;  hence  the  total  number  of  particles  is 
1  X  1015.  The  surface  of  a  single  particle  is  6  X  (0.1  M)*,  or 
6  X  10-10  cm.2,  and  the  total  surface  is  6  X  105  cm.2  The  surface 
tension  of  water  at  17°  C.  is  71  dynes;  hence  the  total  surface 
energy  is  71  X  6  X  105  =  4.32  X  107  ergs. 

This  enormous  figure  shows  that  where  the  surface  of  the  dis- 
perse phase  is  highly  developed,  as  it  is  in  colloidal  solutions,  the 


232  THEORETICAL  CHEMISTRY 

surface  energy  becomes  a  very  important  factor  in  determining 
the  behavior  of  the  system.  This  is  especially  the  case  when 
the  degree  of  aggregation  of  the  colloid  particles  is  changed,  since 
a  relatively  small  change  in  the  amount  of  aggregation  may 
involve  a  great  change  in  the  surface  exposed  and  a  consequent 
change  hi  the  surface  energy. 

The  electrical  and  surface  factors  of  colloidal  solutions  are  not 
independent,  a  very  close  connection  existing  between  them. 

Adsorption.  Solids  possess  the  property  of  condensing  gases, 
liquids  or  dissolved  solids  on  their  surfaces.  This  phenomenon 
is  termed  adsorption,  it  being  a  manifestation  of  the  force  of  adhe- 
sion. It  must  be  remembered  that  the  term  refers  to  surface  con- 
densation only,  since  if  gases  or  liquids  penetrate  the  pores  of  the 
solid,  they  are  said  to  be  occluded  or  absorbed.  Reference  has 
already  been  made  to  the  observations  of  Linder  and  Picton  on 
the  adsorption  of  barium  by  the  hydrogel  of  arsenious  sulphide. 
In  a  similar  manner,  albuminous  substances  take  up  dyes  and 
salts  when  coagulated.  The  adsorptive  tendencies  exhibited  by 
hydrogels  may  be  ascribed  to  their  great  surface  development. 
Many  interesting  examples  of  adsorption  are  furnished  by  dyes, 
some  of  which  are  crystalloidal  while  others  are  colloidal.  Thus 
the  dyeing  of  silk  by  picric  acid,  and  other  dyes,  appears  to  be 
essentially  an  adsorptive  phenomenon.  Furthermore,  the  equi- 
librium between  colloids  and  ions  in  living  cells  is  now  looked  upon 
as  an  adsorption  equilibrium.  The  equilibrium  established  be- 
tween a  solid  and  a  solution,  termed  the  adsorption  equilibrium,  is 
reversible  and  is  characterized  by  the  fact  that  the  amount  of 
dissolved  substance  adsorbed  by  the  solid  increases  much  more 
slowly  than  the  concentration  of  the  solution.  From  this  it 
follows,  that  the  removal  of  the  dissolved  substance  is  relatively 
more  complete  in  dilute  than  in  concentrated  solutions.  It  has 
been  shown  by  Freundlich,  that  if  c8  and  ct  represent  the  con- 
centrations of  the  dissolved  substance  in  the  solid  and  in  the  solu- 
tion respectively,  then  the  distribution  of  the  dissolved  substance 
between  solid  and  solution  may  be  expressed  by  the  exponential 
formula 


COLLOIDAL  SOLUTIONS  233 

where  0  and  p  are  constants,  the  value  of  the  latter  being  greater 
than  unity.  When  the  surface  of  a  solid  is  large  in  comparison 
with  its  volume,  the  conditions  are  favorable  for  adsorption,  but 
whether  this  will  occur  or  not,  depends  upon  the  nature  of  the 
dissolved  substance,  and  especially  upon  the  difference  of  potential 
between  it  and  the  surrounding  medium. 

Preparation  of  Colloidal  Solutions.  Since  1861  when  Graham 
published  his  first  paper  on  colloids,  numerous  investigators  have 
devised  methods  for  the  preparation  of  colloidal  solutions.  Within 
recent  years  our  knowledge  of  this  class  of  solutions  has  been 
greatly  increased,  many  crystalloidal  substances  having  been 
obtained  in  the  colloidal  condition.  As  a  result  of  these  investi- 
gations, we  no  longer  speak  of  crystalloidal  and  colloidal  matter, 
but  use  the  terms,  crystalloid  and  colloid,  to  distinguish  two 
different  states.  In  fact  it  is  now  recognized  that  it  is  simply  a 
matter  of  overcoming  certain  experimental  difficulties,  before  it 
will  be  possible  to  obtain  all  forms  of  matter  in  the  colloidal 
state.  The  scope  of  this  book  forbids  a  detailed  account  of  the 
various  methods  which  have  been  devised  for  the  preparation  of 
colloidal  solutions.*  We  must  content  ourselves  with  a  general 
classification  of  these  methods  into  two  groups  as  follows:  — 
(1)  Chemical  Methods  and  (2)  Electrical  Methods. 

Chemical  Methods.  We  will  give  but  two  typical  examples  of 
chemical  methods  of  preparation  of  colloidal  solutions.  A  col- 
loidal solution  almost  always  results  when  two  compounds  which 
react  to  form  a  precipitate  in  the  presence  of  an  electrolyte  are 
brought  together  in  the  absence  of  an  electrolyte.  Thus,  when 
hydrogen  sulphide  is  passed  into  a  neutral  aqueous  solution  of 
arsenious  oxide  no  precipitate  forms,  but  the  liquid  gives  by 
transmitted  light  a  reddish-yellow  color,  and  by  reflected  light 
exhibits  a  deep  yellowish-red  fluorescence.  The  liquid  contains 
colloidal  arsenious  sulphide.  If  the  experiment  is  repeated  with 
a  slightly  acid  solution  of  arsenious  oxide,  a  yellow  precipitate  of 
arsenious  sulphide  is  formed  immediately. 

Another  chemical  method  for  the  preparation  of  colloidal  solu- 

*  See  "Die  Methoden  zur  Herstellung  Kolloider  Losungen  anorganischer 
Stoffe, "  by  The  Svedberg,  Dresden,  1909. 


234  THEORETICAL  CHEMISTRY 

tions  is  that  devised  by  Gutbier  *  involving  the  use  of  certain 
reducing  agents.  For  example,  a  solution  of  colloidal  gold  may  be 
prepared  in  the  following  manner:  —  1  gram  of  gold  chloride  is 
dissolved  in  1  liter  of  pure  distilled  water,  and  neutralized  with  a 
dilute  solution  of  sodium  carbonate.  To  this  solution  a  very 
dilute,  cold  solution  of  hydrazine  hydrate  (1  :  4000)  is  added.  As 
soon  as  the  first  drop  enters  the  solution  reduction  commences 
with  the  production  of  a  dark  blue  color,  and  after  a  few  cubic 
centimeters  have  been  added  the  reduction  is  practically  complete. 

The  color  of  the  solution  is  almost  indigo  blue,  and  after  dialysis, 
the  colloidal  solution  may  be  preserved  for  an  indefinite  time. 
Filtration  through  paper  has  no  effect  upon  the  solution,  but  the 
addition  of  electrolytes  causes  precipitation.  The  red  colloidal 
gold  may  be  obtained  by  the  addition  of  hydroxylamine  hydro- 
chloride. 

Electrical  Methods.  These  methods  of  preparing  colloidal  solu- 
tions depend  on  the  pulverizing  action  of  a  powerful  electric 
discharge  upon  compact  metals.  In  1897  Bredig  f  discovered, 
while  studying  the  action  of  the  electric  current  on  different 
liquids,  that  if  an  arc  be  established  between  two  metallic  wires 
immersed  in  a  liquid,  minute  particles  of  metal  are  torn  off  from 
the  negative  terminal  and  remain  suspended  in  the  liquid  indefin- 
itely. In  order  to  prepare  a  colloidal  solution  by  the  method  of 
electrical  pulverization,  Bredig  recommends  that  a  direct  current 
arc  be  established  between  wires  of  the  metal  of  which  a  colloidal 
solution  is  desired,  the  ends  of  the  wires  being  submerged  in  water 
in  a  well-cooled  vessel,  as  shown  in  Fig.  67.  A  current  is  used 
the  strength  of  which  ranges  from  5  to  10  amperes,  and  the  volt- 
age of  which  lies  between  30  and  110  volts.  A  rheostat  together 
with  an  ammeter  and  a  voltmeter  are  included  in  the  circuit. 

The  wires  are  brought  in  contact  for  an  instant  in  order  to 
establish  the  arc,  after  which  they  are  separated  about  2  mm. 
During  the  gentle  hissing  of  the  arc,  clouds  of  colloidal  metal  are 
projected  out  into  the  water  from  the  negative  wire,  a  portion  of 
the  metal  torn  off  being  distributed  through  the  water  as  a  coarse 

*  Jour,  prakt.  Chem.,  71,  452  (1905). 

t  Zeit.  Elektrochem.,  4,  514  (1897);  Zeit.  phys.  Chem.,  31,  258  (1899). 


COLLOIDAL  SOLUTIONS 


235 


suspension.  The  size  of  the  particles  disrupted  from  the  negative 
terminal  is  dependent  upon  the  strength  of  the  current,  a  current 
of  10  amperes  producing  a  greater  proportion  of  colloidal  metal 
than  a  current  of  5  amperes.  The  addition  of  a  trace  of  potassium 


Fig.  67. 

hydroxide  to  the  water  has  been  shown  to  facilitate  the  process  of 
pulverization.  When  gold  wires  are  used,  deep  red  colloidal  solu- 
tions are  obtained,  which  after  standing  for  several  weeks  acquire 
a  bluish-violet  color.  With  extra  precautions,  the  red  colloidal 
gold  solutions  may  be  preserved  for  two  years.  These  solutions 
have  been  shown  by  Bredig  to  contain  about  14  mg.  of  gold  per 
100  cc.  In  this  manner  Bredig  prepared  colloidal  solutions  of 
platinum,  palladium,  iridium,  and  silver.  The  method  of  Bredig 
has  been  improved  and  extended  by  Svedberg.*  Svedberg's 
apparatus  is  too  complicated  for  detailed  description  here.  It 
must  suffice  to  state,  that  a  powerful  oscillating  current  is  the 
disruptive  agent,  the  metal  being  used  in  the  form  of  foil.  With 
this  apparatus  he  has  succeeded  in  preparing  colloidal  solutions 
of  tin,  gold,  silver,  copper,  lead,  zinc,  cadmium,  carbon,  silicon, 
selenium,  and  tellurium.  He  has  also  obtained  all  of  the  alkali 
metals  in  the  colloidal  state,  ethyl  ether  being  used  as  the  dis- 
persing medium.  An  interesting  observation  made  by  Svedberg 
in  the  course  of  his  experiments  is  that  the  color  of  a  metal  is 
the  same  in  both  the  colloidal  and  gaseous  states. 

*  Loc.  cit. 


CHAPTER  XI. 
THERMOCHEMISTRY. 

General  Introduction.  A  chemical  reaction  is  almost  invari- 
ably accompanied  by  a  thermal  change.  In  the  majority  of 
cases  heat  is  evolved;  a  violent  reaction  developing  a  large  amount 
of  heat,  while  a  feeble  reaction  develops  a  comparatively  small 
amount.  Such  reactions  are  said  to  be  exothermic.  A  relatively 
small  number  of  chemical  reactions  are  known  which  take  place 
with  an  absorption  of  heat.  These  are  termed  endothermic  reac- 
tions. Instances  of  chemical  reactions  unaccompanied  by  any 
thermal  change  are  very  rare  and  are  almost  wholly  confined  to 
the  reciprocal  transformations  of  optical  isomers.  These  facts, 
which  were  first  observed  by  Boyle  and  Lavoisier,  led  to  the  view 
that  the  amount  of  heat  evolved  in  a  chemical  reaction  might  be 
taken  as  a  measure  of  the  chemical  affinity  of  the  reacting  sub- 
stances. However,  with  the  advance  of  our  theoretical  knowledge, 
it  is  now  known  that  this  is  not  true,  although  a  parallelism 
between  heat  evolution  and  chemical  affinity  frequently  exists. 

Thermochemistry  is  concerned  with  the  thermal  changes  which 
accompany  chemical  reactions. 

Thermal  Units.  Heat  is  a  form  of  energy,  and  like  other 
forms  of  energy  it  may  be  resolved  into  two  factors;  an  intensity 
factor,  the  temperature,  and  a  capacity  factor,  which  may  be 
measured  in  any  one  of  several  units.  Among  these  units  those 
defined  below  are  the  most  frequently  employed. 

The  small  calorie  (cal.)  is  the  quantity  of  heat  required  to  raise 
the  temperature  of  1  gram  of  water  from  15°  C.  to  16°  C.  The 
temperature  interval  is  specified  because  the  specific  heat  of  water 
varies  with  the  temperature.  The  large  or  kilogram  calorie  (Cal.) 
is  the  quantity  of  heat  required  to  raise  the  temperature  of  1000 
grams  of  water  from  15°  C.  to  16°  C.  The  Ostwald  or  average 

236 


THERMOCHEMISTRY  237 

calorie  (K),  is  the  quantity  of  heat  required  to  raise  the  temper- 
ature of  1  gram  of  water  from  the  melting  point  of  ice  to  the 
boiling  point  of  water  under  a  pressure  of  760  mm.  of  mercury. 
It  is  approximately  equal  to  100  cal.  or  to  0.1  Cal.  The  joule  (j), 
a  unit  based  on  the  C.G.S.  system,  is  equal  to  107  ergs.  This 
being  inconveniently  small  is  generally  multipled  by  1000,  giving 
the  kilojoule  (J),  which  is  therefore  equal  to  1010  ergs.  The  last 
two  units  are  open  to  the  objection  that  their  values  are  depend- 
ent upon  the  mechanical  equivalent  of  heat,  any  change  in  the 
accepted  value  of  which  would  involve  a  correction  of  the  unit  of 
heat.  The  different  capacity  factors  of  heat  energy  are  related 
as  follows :  — 
1  cal.  =  0.001  Cal.  =  0.01  K  (approx.)  =  4.183  j  =  0.004183  J. 

Thermochemical  Equations.  In  order  to  represent  the  changes 
in  energy  which  accompany  chemical  reactions,  an  additional 
meaning  has  been  assigned  to  the  chemical  symbols.  As  ordina- 
rily used,  these  symbols  represent  only  the  molecular  or  formula 
weights  of  the  reacting  substances.  In  a  thermochemical  or 
energy  equation  the  symbols  represent  not  only  the  weight  in 
grams  expressed  by  the  formula  weights  of  the  substances,  but 
also  the  amount  of  heat  energy  contained  in  the  formula  weight 
in  one  state  as  compared  with  the  energy  contained  in  a  standard 
state.  For  example,  the  energy  equation, 

C  +  2  O  =  C02  +  94,300  cal., 

indicates  that  the  energy  contained  in  12  grams  of  carbon  and 
32  grams  of  oxygen  exceeds  the  energy  contained  in  44  grams  of 
carbon  dioxide,  at  the  same  temperature,  by  94,300  calories.  In 
writing  energy  equations  it  is  very  essential  that  we  have  some 
means  of  distinguishing  between  the  different  states  of  aggrega- 
tion of  the  reacting  substances,  since  the  energy  content  of  a 
substance  is  not  the  same  in  the  gaseous,  liquid,  and  solid  states. 
In  the  system  proposed  by  Ostwald,  ordinary  type  is  used  for 
liquids,  heavy  type  for  solids,  and  italics  for  gases.  Another  and 
more  convenient  system  has  been  proposed,  in  which  solids  are 
designated  by  enclosing  the  symbol  or  formula  within  square 
brackets;  liquids  by  the  simple,  unbracketed  symbol  or  formula; 


238 


THEORETICAL  CHEMISTRY 


and  gases  by  enclosing  the  symbol  or  formula  within  parentheses. 

The  above  equation  should,  therefore,  be  written  in  the  following 

manner :  — 

[C]  +  (2  0)  =  (C02)  +  94,300  cal. 

Thermochemical  Measurements.     In  order  to  measure  the 

number  of  calories  evolved  or  ab- 
sorbed when  substances  react,  it  is 
necessary  that  the  reaction  should 
proceed  rapidly  to  completion.  This 
condition  is  fulfilled  by  two  classes 
of  processes.  In  the  first  class  we 
may  mention  the  processes  of  solu- 
tion, hydration,  and  neutralization; 
and  in  the  second  class,  the  process 
^  of  combustion. 

The  apparatus  used  for  measur- 
ing the  capacity  factor  of  heat  energy 
is  a  calorimeter.  This  instrument 
may  be  given  a  variety  of  forms, 
depending  upon  the  particular  use 
to  which  it  is  to  be  put.  A  simple 
form  of  calorimeter  is  shown  in  Fig. 
68.  It  consists  of  two  concentric 
metal  cylinders,  A  and  B}  insulated 
from  each  other  by  an  air  jacket, 
the  inner  vessel  being  suppported 
on  vulcanite  points.  Through  a 
vulanite  cover  passes  a  thin  walled 
test  tube,  in  which  the  reaction  is 


Fig.  68. 


allowed  to  take  place.  An  accurate  thermometer  and  a  ring-stirrer 
also  pass  through  the  cover  of  the  calorimeter.  In  order  to  deter- 
mine the  thermal  capacity  of  the  calorimeter,  B  is  nearly  filled  with 
water,  and  a  known  mass  of  water,  ra,  at  a  temperature  ti  is  intro- 
duced into  (7.  Let  the  initial  temperature  of  the  water  in  B  be  ^.  The 
water  in  B  is  stirred  until  the  contents  of  both  B  and  C  have  acquired 
the  same  temperature,  U.  When  thermal  equilibrium  has  been  estab- 
lished, it  is  evident  that  m  (t\  —  £3)  calories  are  required  to  raise 


THERMOCHEMISTRY  239 

the  temperature  of  the  apparatus  and  the  water  in  B,  (^  —  £3) 
degrees.  From  this  data  it  is  an  easy  matter  to  calculate  the 
number  of  calories  required  to  raise  the  temperature  of  the  appar- 
atus and  water  in  B  1  degree,  this  being  the  thermal  capacity  of 
the  apparatus.  The  calorimeter  may  now  be  used  to  determine 
the  heat  evolved  or  absorbed  hi  a  reaction.  Suppose,  for  exam- 
ple, that  it  is  desired  to  measure  the  heat  of  neutralization  of  an 
acid  by  a  base.  Equivalent  quantities  of  both  acid  and  base  are 
dissolved  in  equal  volumes  of  water,  care  being  taken  to  make  the 
solutions  dilute.  A  definite  volume  of  one  solution  is  introduced 
into  C  and  an  equal  volume  of  the  other  solution  is  placed  in  a 
vessel  from  which  it  can  be  quickly  and  completely  transferred  to 
C.  When  both  solutions  have  acquired  the  same  temperature, 
the  thermometer  in  B  is  read  and  then  the  two  solutions  are 
mixed.  When  the  reaction  is  complete,  the  temperature  of  the 
water  in  B  is  again  noted.  If  the  thermal  capacity  of  the  calori- 
meter is  Q,  and  the  rise  hi  temperature  produced  by  the  reaction 
is  B,  then  Qd  is  the  amount  of  heat  evolved  by  the  reaction.  To 
this  quantity  of  heat  must  be  added  the  number  of  calories  re- 
quired to  raise  the  temperature  of  the  products  of  the  reaction  6 
degrees.  The  solutions  of  the  products  being  dilute,  their  specific 
heats  may  be  assumed  to  be  equal  to  unity.  From  the  total  quan- 
tity of  heat  so  obtained,  the  number  of  calories  evolved  when  mo- 
lecular quantities  react  can  be  readily  calculated.  The  chief  source 
of  error  in  calorimetric  measurements  is  loss  by  radiation.  This 
may  be  reduced  to  a  minimum  (1)  by  making  the  thermal  capac- 
ity of  the  calorimeter  large,  and  (2)  by  so  arranging  matters  that 
the  initial  temperature  of  the  water  in  the  calorimeter  is  as  much 
below  the  temperature  of  the  room  as  the  final  temperature  is 
above  it. 

The  Combustion  Calorimeter.  The  combustion  of  many  sub- 
stances, such  as  organic  compounds,  proceeds  very  slowly  in  air 
under  ordinary  pressures.  Such  reactions  can  be  accelerated,  if 
they  are  caused  to  take  place  in  an  atmosphere  of  compressed 
oxygen.  For  this  purpose  the  combustion  calorimeter  was  de- 
vised by  Berthelot.*  In  this  apparatus  the  essential  feature  is 
*  Ann.  Chim.  Phys.,  (5),  23,  160  (1881);  (6),  10,  433  (1887). 


240 


THEORETICAL  CHEMISTRY 


the  so-called  combustion  bomb,  shown  in  Fig.  69.  This  consists 
of  a  strong  steel  cylinder  lined  with  platinum  or  gold,  and  fur- 
nished with  a  heavy  threaded  cover.  The  substance  to  be 
burned  is  placed  in  a  platinum  capsule  fastened  to  the  support 
R,  and  a  short  piece  of  fine  iron  wire  of  known  mass  is  connected 
with  the  electric  terminals  Z,  Z,  the  middle  portion  of  the  wire 
dipping  into  the  substance.  The  cover  is  then  screwed  down 
tight,  and  the  bomb  is  filled  with  oxygen  under  a  pressure  of  from 
20  to  25  atmospheres.  The  bomb  is  then 
submerged  in  the  calorimeter,  as  shown  in 
Fig.  70.  The  mass  of  water  in  the  calorim- 
eter being  known  and  its  temperature  having 
been  read,  an  electric  current  is  passed  through 
S  the  iron  wire  in  the  bomb  causing  it  to  burn 
and  thus  ignite  the  substance.  The  rise  in 
temperature  due  to  the  combustion  is  ob- 
served, and  the  quantity  of  heat  evolved  is 
calculated.  Corrections  must  be  applied  for 
loss  by  radiation,  for  the  heat  evolved  from 
the  combustion  of  the  iron,  and  for  the  heat 
evolved  from  the  oxidation  of  the  nitrogen  of 
the  residual  air  in  the  bomb. 

For  the  details  of  the  method  of  determin- 
ing heats  of  combustion  the  student  must 
consult  a  laboratory  manual. 
Law  of  Lavoisier  and  Laplace.  In  1 780,  Lavoisier  and  Laplace,* 
as  a  result  of  their  thermochemical  investigations,  enunciated  the 
following  law :  —  The  quantity  of  heat  which  is  required  to  decompose 
a  chemical  compound  is  precisely  equal  to  that  which  was  evolved  in 
the  formation  of  the  compound  from  its  elements.  This  first  law  of 
thermochemistry  will  be  seen  to  be  a  direct  corollary  of  the  law 
of  the  conservation  of  energy  which  was  first  clearly  stated  by 
Mayer  in  1842. 

Law  of  Constant  Heat  Summation.  A  generalization  of  funda- 
mental importance  to  the  science  of  thermochemistry  was  discov- 
ered in  1840  by  Hess.f  He  pointed  out  that  the  heat  evolved  in  a 

*  Oeuvres  de  Lavoisier,  Vol.  II,  p.  283. 
t  Pogg.  Ann.,  50,  385  (1840). 


Fig.  69. 


THERMOCHEMISTRY 


241 


chemical  process  is  the  same  whether  it  takes  place  in  one  or  in  several 
steps.     This  is  known  as  the  law  of  constant  heat  summation.     The 


Fig.  70. 

truth  of  the  law  may  be  illustrated  by  the  equality  of  the  heat  of 
formation  of  ammonium  chloride  in  aqueous  solution,  when  pre- 
pared in  two  different  ways. 


242  THEORETICAL  CHEMISTRY 

Thus, 

(A) 


(NH3)  +  (HC1)  =  [HN4C1]  +42,100  cal. 

[NH4C1]  +  aq.    =  NH4C1,  aq.        -  3,900  cal. 

38,200  cal. 

(B) 


(NH3)  +  aq.  =  NH3,  aq.  +  8,400  cal. 

(HC1)  +  aq.  =  HC1,  aq.  +17,300  cal. 

NH3,  aq.  +  HC1,  aq.  =  NH4Cl,aq.  +12,300  cal. 

38,000  cal. 

It  will  be  observed  that  the  total  amount  of  heat  evolved  in 
the  formation  and  solution  of  ammonium  chloride  is  the  same 
within  the  limits  of  experimental  error,  whether  gaseous  ammonia 
and  hydrochloric  acid  are  allowed  to  react  and  the  resulting  prod- 
uct is  dissolved  in  water,  or  whether  the  gases  are  each  dissolved 
separately  and  then  allowed  to  react.  It  should  be  noted  that 
when  a  substance  is  dissolved  in  so  much  water  that  the  addition 
of  more  water  or  the  removal  of  a  small  portion  of  water  produces 
no  thermal  effect,  it  is  customary  to  denote  it  by  the  symbol  aq. 
(Latin  aqua  =  water).  Thus, 

NH4C1,  aq.  +  nH2O  =  NH4C1,  aq., 
NH4C1,  aq.  -  nH20  =  NH4C1,  aq. 

By  means  of  the  law  of  constant  heat  summation  it  is  possible  to 
find  indirectly  the  amount  of  heat  developed  or  absorbed  by  any 
reaction,  even  though  it  is  impossible  to  carry  it  out  experimen- 
tally. For  example,  it  is  impossible  to  measure  the  heat  evolved 
when  carbon  burns  to  carbon  monoxide.  But  the  heat  evolved 
when  carbon  monoxide  burns  to  carbon  dioxide,  and  also  the  heat 
evolved  when  carbon  burns  to  carbon  dioxide,  can  be  accurately 
determined.  The  energy  equations  are  as  follows:  — 

[C]  +  2(0)  =  (C02)  +  94,300  cal.  (1) 

(CO)  +  (0)  =  (C02)  +  67,700  cal.  (2) 


THERMOCHEMISTRY  243 

Treating  these  equations  algebraically,  and  subtracting  equation 
(2)  from  equation  (1),  we  have 

[C]  +  (0)  =  (CO)  +  26,600  cal., 

or,  the  heat  of  combustion  of  carbon  to  carbon  monoxide  is  26,600 
calories.  Again,  as  a  further  illustration  of  the  applicability  of 
the  law  of  Hess,  we  may  take  the  calculation  of  the  heat  of  forma- 
tion of  hydriodic  acid  from  its  elements,  making  use  of  the  follow- 
ing energy  equations :  — 

2  KI,  aq.  +  2  (Cl)  =  2  KC1,  aq.  +  2  [I]  +  524  K  (1) 

2  HI,  aq.  +  2  KOH,  aq.  =  2  KI,  aq.   +  2  (H20)  +  274  K  (2) 

2  HC1,  aq.  +  2  KOH,aq.  =  2  KC1,  aq.  +  2  (H20)  +  274  K  (3) 

2  (HI)  +  aq.  =  2  HI,  aq.  +  384  K,  (4) 

2  (HC1)  +  aq.  =  2  HC1,  aq.  +  346  K,  (5) 

2  (H)  +  2  (Cl)  =  2  (HC1)  +  440  K,  (6) 

adding  equations  (1)  and  (2), 

2  (Cl)  +  2  HI,  aq.  +  2  KOH,  aq.  =  2  KC1,  aq.  +  2  [I]  +  2  (H20) 

+  798K.  (7) 

Subtracting  equation  (3)  from  equation  (7), 

2  (Cl)  +  2  HI,  aq.  -  2  HC1,  aq.  =  2  [I]  +  524  K, 
or 

2  (Cl)  -f  2  HI,  aq.  =  2  [I]  +  2  HC1,  aq.  +  524  K,  (8) 

adding  equations  (4)  and  (8), 

2  (HI)  +  aq.  +  2  (Cl)  =  2  [I]  -f  2  HC1,  aq.  +  908  K,        (9) 
subtracting  equation  (5)  from  equation  (9), 

2  (HI)  +  2  (Cl)  -  2  (HC1)  =  2  [I]  +  562  K, 
or 

2  (HI)  +  2  (Cl)  =  2  [I]  +  2  (HC1)  +  562  K,  (10) 

subtracting  equation  (6)  from  equation  (10), 

2  (HI)  -  2  (H)  =  2  [I]  +  122  K, 
or 

2(H)  +  2(I)  =  2  (HI)- 122  K. 

In  a  similar  manner,  practically  any  heat  of  formation  may  be 
calculated,  provided  the  proper  energy  equations  are  combined. 


244  THEORETICAL  CHEMISTRY 

Heat  of  Formation.  The  intrinsic  energy  of  the  substances 
entering  into  chemical  reaction  is  unknown,  the  amount  of  heat 
evolved  or  absorbed  in  the  process  being  simply  a  measure  of  the 
difference  between  the  energy  of  the  reacting  substances  and  the 
energy  of  the  products  of  the  reaction.  Thus,  in  the  equation 

[C]  -f  2  (0)  =  (C02)  +  94,300  cal, 

the  difference  between  the  energy  of  a  mixture  of  12  grams  of 
carbon  and  32  grams  of  oxygen,  and  the  energy  of  44  grams  of 
carbon  dioxide  is  seen  to  be  94,300  calories.  The  equation  is 
clearly  incomplete  since  we  have  no  means  of  determining  the 
intrinsic  energies  of  free  carbon  and  oxygen.  Furthermore,  since 
the  elements  are  not  mutually  convertible,  we  have  no  means  of 
determining  the  difference  in  energy  between  them.  It  is  cus- 
tomary, therefore,  in  view  of  this  lack  of  knowledge,  to  put  the 
intrinsic  energies  of  the  elements  equal  to  zero. 

If  the  heats  of  formation  of  the  substances  present  in  a  reac- 
tion are  known,  it  is  much  simpler  to  substitute  these  in  the 
energy  equation  and  solve  for  the  unknown  term.  This  method 
avoids  the  laborious  process  of  elimination  from  a  large  number  of 
energy  equations,  as  in  the  preceding  pages.  If  all  of  the  sub- 
stances involved  in  a  reaction  are  considered  as  decomposed  into 
their  elements,  it  is  evident  that  the  final  result  of  the  reaction 
will  be  the  difference  in  the  sums  of  the  heats  of  formation  on  the 
two  sides  of  the  equation.  This  leads  to  the  following  rule:  — 
To  find  the  quantity  of  heat  evolved  or  absorbed  in  a  chemical  reac- 
tion, subtract  the  sum  of  the  heats  of  formation  of  the  substances 
initially  present  from  the  sum  of  the  heats  of  formation  of  the  products 
of  the  reaction,  placing  the  heat  of  formation  of  all  elements  equal  to 
zero. 

The  energy  equation  for  the  formation  of  carbon  dioxide  from 
its  elements  may  then  be  written  as  follows :  — 
0  +  0  =  (C02)  +  94,300  cal., 

(CO,)  =  -  94,300  cal. 

That  is,  the  energy  of  1  mol  of  carbon  dioxide  is  —94,300  calories. 
Therefore  in  writing  an  energy  equation  we  make  use  of  the  fol- 


THERMOCHEMISTRY  245 

lowing  rule :  —  Replace  the  formulas  of  each  compound  in  the  equa- 
tion representing  the  reaction  by  the  negative  values  of  their  respective 
heats  of  formation  and  solve  for  the  unknown  term.  This  unknown 
term  may  be  either  the  heat  of  a  reaction  or  the  heat  of  formation 
of  one  of  the  reacting  substances.  The  following  examples  will 
serve  to  illustrate  the  application  of  the  above  rules :  — 

(1)  Let  it  be  required  to  find  the  heat  of  the  following  reaction 

[MgClJ  +  2  [Na]  =  2  [NaCl]  +  [Mg]  +  x, 

where  x  is  the  heat  of  the  reaction.  The  heat  of  formation  of 
MgCl2  is  151  CaL,  and  that  of  NaCl  is  97.9  Cal.,  therefore, 

-  151  +  0  =  -  (2  X  97.9)  +  0  +  x, 
or 

x  =  44.8  Cal. 

(2)  The  heat  of  combustion  of  1  mol  of  methane  is  213.8  Cal., 
and  the  heats  of  formation  of  the  products,  carbon  dioxide  and 
water,  are  94.3  Cal.  and  68.3  Cal.,  respectively.     Let  it  be  required 
to  find  the  heat  of  formation  of  methane.     Representing  the  heat 
of  formation  of  methane  by  x,  we  have 

(CH4)  +  2  (02)  =  (CO,)  +  2  (H20)  +  213.8  Cal., 

-  x  +  0  =  -(94.3  +  2  X  68.3)+  213.8, 
or 

x  =  17.1  Cal. 

(3)  The  heat  of  combustion  of  1  mol  of  carbon  disulphide  is 
265.1  CaL,  the  thermochemical  equation  being 

CS2  +  3  (Os)  =  (CO,)  +  2  (SO,)  +  265.1  Cal. 

The  heats  of  formation  of  carbon  dioxide  and  sulphur  dioxide  are 
94.3  Cal.  and  71  Cal.  respectively.  The  heat  of  formation  of 
carbon  disulphide  x,  may  then  be  calculated  as  follows :  - — 

-  x  +  o  =  -  94.3  -  2  X  71  +  265.1, 
or 

x  =  -28.8  Cal. 

Carbon  disulphide  is  thus  seen  to  be  an  endothermic  compound. 

Heat  of  Solution.  The  thermal  change  accompanying  the 
solution  of  1  mol  of  a  substance  in  so  large  a  volume  of  solvent 
that  subsequent  dilution  of  the  solution  causes  no  further  thermal 


246 


THEORETICAL  CHEMISTRY 


change  is  termed  the  heat  of  solution.  The  solution  of  neutral 
salts  is  generally  an  endothermic  process.  This  fact  may  be 
readily  accounted  for  on  the  hypothesis  that  considerable  heat 

HEATS  OF  FORMATION  AND  SOLUTION. 


Substance. 


Heat9f 
Formation. 


Heat  of 
Solution. 


Water,  vapor 58 . 7 

Water,  liquid 68.4 

Hydrochloric  acid 22 . 0 

Sulphuric  acid 193 . 1 

Ammonia 12.0 

Nitric  acid 41 . 9 

Phosphoric  acid 302.9 

Potassium  hydroxide 103 . 2 

Potassium  chloride 104 . 3 

Potassium  bromide 95 . 1 

Potassium  iodide 80 . 1 

Potassium  nitrate 119.5 

Sodium  hydroxide 101 . 9 

Sodium  chloride 97 . 6 

Sodium  bromide 85 . 6 

Sodium  sulphate 328.8 

Sodium  nitrate 111.3 

Sodium  carbonate 272 . 6 

Ammonium  chloride 75 . 8 

Ammonium  nitrate 88 . 0 

Calcium  hydroxide 215 . 0 

Calcium  chloride 170.0 

Magnesium  sulphate 502 . 0 

Ferrous  chloride 82 . 0 

Ferric  chloride 96 . 1 

Zinc  chloride 97 . 0 

Zinc  sulphate 30.0 

Cadmium  chloride 293.2 

Cupric  chloride 51.6 

Cupric  sulphate 182 . 6 

Mercuric  chloride 53 . 2 

Silver  nitrate 28.7 

Stannous  chloride 80. 8 

Stannic  chloride 127.3 

Lead  chloride 82.8 

Lead  nitrate . .  105 . 5 


20.3 

17.8 

8.4 

7.2 

2.7 

13.3 

-3.1 

-5.1 

-5.1 

-8.5 

10.9 

1.2 

-0.2 

0.2 

-5.0 

5.6 

-4.0 

-6.2 

3.0 

17.4 

20.3 

17.9 

63.3 

15.6 

18.5 

3.0 

11.1 

15.8 

-3.3 

-5.4 

0.3 

29.9 

-6.8 

-7.6 


must  be  absorbed  as  heat  of  fusion  and  heat  of  vaporization  before 
the  solid  salt  can  assume  a  condition  in  solution  which  closely 
resembles  that  of  a  gas.  The  heat  of  solution  of  hydrated  salts 
is  less  than  the  heat  of  solution  of  the  corresponding  anhydrous 


THERMOCHEMISTRY 


247 


salts.  For  example,  the  heat  of  solution  of  1  mol  of  anhydrous 
calcium  nitrate  is  4000  calories,  while  the  heat  of  solution  of  1  mol 
of  the  tetrahydrate  is  —7600  calories.  The  difference  between 
the  heats  of  solution  of  the  anhydrous  and  hydrated  salts  is  termed 
the  heat  of  hydration.  The  heats  of  formation  and  heats  of  solu- 
tion in  water  of  some  of  the  more  common  compounds  are  given 
in  the  preceding  table,  the  values  being  expressed  in  large  calories. 

Heat  of  Dilution.  The  heat  of  dilution  of  a  solution  is  the 
quantity  of  heat  per  mol  of  solute  which  is  evolved  or  absorbed 
when  the  solution  is  greatly  diluted.  Beyond  a  certain  dilution, 
further  addition  of  solvent  produces  no  thermal  change.  While 
there  is  a  definite  heat  of  solution  for  a  particular  solute  in  a  par- 
ticular solvent,  the  heat  of  dilution  remains  indefinite,  since  the 
latter  is  dependent  upon  the  degree  of  dilution.  Those  gases 
which  obey  Henry's  law  are  practically  the  only  substances  which 
have  no  appreciable  heats  of  solution  or  dilution. 

The  following  tables  give  the  heats  of  dilution  of  hydrochloric 
and  nitric  acids. 


HEAT  OF  DILUTION  OF  SOLUTIONS  OF  HYDROCHLORIC 

ACID. 

Heat  of  solution  =  20.3  cal. 


HC1+H20.. 
HC1+2H2O.. 

5.37 
11.36 

HC1+10  H2O 

16  16 

HC1+50  H2O     . 

17  1 

HC1+300H2O.  .  . 

17  3 

HEAT  OF  DILUTION  OF  SOLUTIONS  OF  NITRIC  ACID. 
Heat  of  solution  =  7.15  cal. 


HNO3+H2O.  . 

3  84 

HNO3+2  H2O  

2.32 

HNO3+4H2O  

1.42 

HNO3+6  H2O 

0  2 

HNO3+  8  H2O 

—0  04 

HNO3+100  H2O    .... 

-0  03 

248  THEORETICAL  CHEMISTRY 

Reactions  at  Constant  Volume.  When  a  chemical  reaction 
takes  place  without  any  change  in  volume,  or  when  the  external 
work  resulting  from  a  change  in  volume  is  not  included  in  the 
heat  of  the  reaction,  the  process  is  said  to  take  place  at  constant 
volume.  That  is  to  say,  the  condition  of  constant  volume  is  a 
condition  which  involves  no  external  work,  either  positive  or 
negative.  Under  these  conditions  the  total  energy  of  the  react- 
ing substances  is  equal  to  the  total  energy  of  the  products  of  the 
reaction,  plus  the  quantity  of  heat  developed  by  the  reaction. 

Reactions  at  Constant  Pressure.  When  a  chemical  reaction 
is  accompanied  by  a  change  in  volume,  the  system  suffers  a  loss 
of  heat  equivalent  to  the  work  done  against  the  atmosphere,  if 
the  volume  increases;  or  the  system  gains  an  amount  of  heat 
equivalent  to  the  work  done  on  the  system  by  the  atmosphere, 
if  the  volume  decreases.  Under  these  conditions  the  reaction  is 
said  to  take  place  at  constant  pressure.  The  difference  between 
constant  volume  and  constant  pressure  conditions,  then,  is  that 
under  the  former,  the  heat  equivalent  of  the  work  corresponding 
to  any  change  in  volume  which  may  occur  is  not  considered  as 
having  any  effect  upon  the  energy  of  the  system;  whereas  under 
the  latter,  due  account  is  taken  of  the  change  in  energy  resulting 
from  change  in  volume.  Suppose  that  in  a  reaction,  1  mol  of  gas 
is  formed.  Under  standard  conditions  of  temperature  and 
pressure  the  volume  of  the  system  will  be  increased  by  22.4  liters. 
The  formation  of j  gas  involves  the  performance  of  work  against 
the  atmosphere,  this  work  being  done  at  the  expense  of  the  heat 
energy  of  the  system.  To  calculate  the  heat  equivalent  of  the 
work  done,  let  us  imagine  the  gas  enclosed  in  a  cylinder  fitted  with 
a  piston  whose  area  is  1  square  centimeter.  The  normal  pressure 
of  the  atmosphere  on  the  piston  is  76  cm.  of  mercury  or  1033.3 
grams  per  square  centimeter.  If  the  increase  in  the  volume  of 
the  gas  is  22.4  liters,  the  piston  will  be  raised  through  22,400  cm. 
and  the  work  done  will  be  1033.3  X  22,400  gram-centimeters. 
The  heat  equivalent  of  this  change  in  volume  will  be  (1033.3  X 
22,400)  -f-  42,600  =  542.3  calories  or  0.5423  large  calories.  This 
amount  of  heat  must  be  added  to  the  heat  of  the  reaction.  It 
should  be  observed  that  this  correction  is  independent  of  the  actual 


THERMOCHEMISTRY  249 

value  of  the  pressure  upon  the  system.  Thus,  if  the  pressure  is 
increased  n  times,  the  volume  of  the  gas  will  be  reduced  to  1/n 
of  its  former  value,  and  the  work  done  will  involve  moving  the 
piston  through  1/n  of  the  distance  against  an  n-fold  pressure, 
which  is  plainly  equivalent  to  the  former  amount  of  work.  While 
the  correction  is  independent  of  the  pressure  it  is  not  independent 
of  the  temperature.  The  familiar  equation,  pv  =  RT,  shows  us 
that  the  work  done  by  a  gas  is  directly  proportional  to  its  absolute 
temperature.  Thus,  if  a  gas  is  evolved  at  273°  absolute,  it  will 
occupy  double  the  volume  it  would  occupy  at  0°,  and  the  work 
done  at  273°  will  involve  moving  the  piston  through  twice  the 
distance  that  it  would  have  to  be  moved  at  0°.  Theoretically,  a 
gas  evolved  at  the  absolute  zero  would  occupy  no  volume  and 
hence  no  work  would  be  done.  Introducing  the  correction  for 
temperature,  we  see  that 

-  XT  =  1.986  T  cal., 

must  be  added  to  the  heat  of  the  reaction,  where  T  is  the  absolute 
temperature  at  which  the  change  in  volume  occurs.  For  all 
ordinary  purposes  it  is  sufficiently  accurate  to  take  2  T  calories 
as  the  correction.  Thus,  suppose  n  mols  of  gas  to  be  formed  in  a 
reaction  at  17°  C.,  the  amount  of  heat  absorbed  will  be 

nX2(273  +  17)  =  580 n  cal. 

Under  constant  pressure  conditions,  the  symbols,  in  addition  to 
their  usual  significance,  represent  the  energy  plus  or  minus  the 
term,  2  T  per  mol,  the  positive  or  negative  sign  being  used 
according  as  the  gas  is  absorbed  or  formed.  Since  the  constant 
volume  condition  is  a  condition  in  which  no  account  is  taken  of  the 
external  work,  even  if  a  change  in  volume  does  occur  during  the 
reaction,  and  the  constant  pressure  condition  is  one  in  which 
the  external  work  is  taken  into  consideration,  it  is  apparent  that 
the  relation  of  the  heat  energy  of  a  reaction  at  constant  volume, 
Qv,  to  the  heat  energy  at  constant  pressure  Qp,  can  be  represented 
by  the  equation 

QP  =  Qv 


250  THEORETICAL  CHEMISTRY 

where  n  denotes  the  number  of  mols  of  gas  formed  in  excess  of 
those  initially  present.  This  equation  is  of  great  importance  in 
connection  with  the  determination  of  heats  of  combustion  in  the 
bomb-calorimeter  in  which  the  reactions  necessarily  take  place 
under  constant  volume  conditions.  Since  it  is  customary  to  state 
heats  of  reaction  under  constant  pressure  conditions,  the  foregoing 
equation  makes  it  possible  to  convert  heats  of  combustion  deter- 
mined under  constant  volume  conditions  into  heats  of  combustion 
under  constant  pressure  conditions.  For  example,  the  combustion 
of  naphthalene  takes  place  in  accordance  with  the  equation 
C10H8  +  12  (02)  =  10  (C02)  +  4  (H20)  +  1242.95  Cal. 

It  is  apparent  that  the  combustion  is  accompanied  by  the  forma- 
tion of  2  mols  of  gas,  and  at  15°  C.  the  correction  will  be 

Qp  =  1242.95  -  2  X  0.002  (273  +  15), 
or 

Qp  =  1241.8  Cal. 

The  volume  occupied  by  solids  or  liquids  is  so  small  as  to  be 
negligible  and  does  not  enter  into  these  calculations. 

Variation  of  Heat  of  Reaction  with  Temperature.  If  a  chem- 
ical reaction  be  allowed  to  take  place  first  at  the  temperature  fa, 
and  then  at  the  temperature  k,  the  amounts  of  heat  developed  in 
the  two  cases  will  be  found  to  be  quite  different.  Let  Qi  and  Q2 
represent  the  quantities  of  heat  evolved  at  the  temperatures  fa 
and  k,  respectively.  Let  us  imagine  that  the  reaction  takes  place 
at  the  temperature  ti,  Qi  units  of  heat  being  evolved;  and  then 
let  the  products  of  the  reaction  be  heated  to  the  temperature  iz. 
If  c'  represents  the  total  thermal  capacity  of  the  products  of  the 
reaction,  then  the  quantity  of  heat  necessary  to  produce  this  rise 
in  temperature  will  be  c'  fe  —  fa).  Now  let  us  imagine  the 
reacting  substances,  at  the  temperature  fa,  to  be  heated  to  the 
temperature  k,  and  then  allowed  to  react  with  the  evolution  of 
Qz  units  of  heat.  The  heat  necessary  to  produce  this  rise  in 
temperature  in  the  reacting  substances  is  c  (k  —  Zi),  where  c 
is  the  total  thermal  capacity  of  the  original  substances.  Having 
started  with  the  same  substances  at  the  same  initial  temperature, 
and  having  obtained  the  same  products  at  the  same  final  temper- 


THERMOCHEMISTRY  251 

ature,  we  have,  according  to  the  law  of  the  conservation  of 
energy, 

Qi  -  c'  &  -  «0  = 
or 


or,  where  the  change  in  temperature  is  very  small, 


If  c'  is  greater  than  c  then  the  sign  of  dQ/dt  will  be  negative,  or,  in 
other  words,  an  increase  in  temperature  will  cause  a  decrease  in 
the  heat  of  reaction.  On  the  other  hand,  if  c  is  greater  than  c', 
dQ/dt  will  be  positive  and  the  heat  of  reaction  will  increase  with 
the  temperature. 

EXAMPLE.     The   reaction  between   hydrogen   and   oxygen   at 
18°  C.  is  represented  by  the  following  equation:  — 

2  (H2)  +  (02)  =  2  (H20)  +  1367.1  K. 

Suppose  it  is  required  to  find  how  much  heat  will  be  evolved  when 
equal  masses  of  the  two  gases  react  at  110°  C.,  the  product  of  the 
reaction  being  maintained  at  this  temperature,  and  the  pressure 
remaining  constant.  The  specific  heats  per  gram  of  the  different 
substances  involved  are  as  follows  :  — 

Hydrogen   =  3.409;    Oxygen  =  0.2175;    Water  (between  18° 
and   100°)  =  1;   Water    (between   100°   and   110°)  =  0.5. 
The  heat  of  vaporization  of  water  is  537  calories  per  gram. 

For  liquid  water  per  degree  we  have, 
dQ/dt  =  (4  X  3.409  +  32  X  0.2175)  -  (36  X  1)  =  -  15.404  cal. 

and  for  (100°-  18°)  =82°,  we  have,  82  X  (-15.404)  =  -1263  cal. 
The  heat  of  formation  of  liquid  water  at  100°  is,  therefore, 
1367.1  -  12.63  =  1354.47  K. 

When  the  liquid  water  is  vaporized  at  100°,  (36  X  537)  calories  of 
this  heat  is  absorbed,  or  the  formation  of  steam  at  100°  from 
hydrogen  and  oxygen,  evolves 

1354.47  -  193.32  =  1161.15  K, 


252 


THEORETICAL  CHEMISTRY 


For  steam  per  degree,  we  have, 

dQ/dt  =  (4  X  3.409  +  32  X  0.2175)  -  (36  X  0.5)  =  2.596  cal., 
and  for  the  interval  (110°  -  100°)  =  10°, 

10  X  2.596  =  25.96  cal. 
Or  for  the  total  heat  evolved,  we  have 

1161.15  +  0.2596  =  1161.41  K. 

Heats  of  Combustion.  The  heat  evolved  during  the  complete 
oxidation  of  unit  mass  of  a  substance  is  termed  its  heat  of 
combustion.  The  unit  of  mass  commonly  chosen  in  all  physico- 
chemical  calculations  is  the  mol.  An  enormous  amount  of 
experimental  work  has  been  done  by  Thomsen,*  Berthelot,f  and 
Langbein  t  on  the  determination  of  the  heats  of  combustion  of  a 
large  number  of  organic  compounds.  A  few  of  their  results  are 
given  in  the  accompanying  tables. 

SATURATED  HYDROCARBONS. 


Hydrocarbon. 

Heat  of 
Combustion. 

Difference. 

Methane   CH4 

Cal. 
211   9 

Cal. 

Ethane  C2H6 

370  4 

158.5 

Propane,  CaHg  .  .               ... 

529  2 

158.8 

Butane,  C4Hio  

687.2 

158.0 

Pentane,  CsH^  

847.1 

159.9 

UNSATURATED  HYDROCARBONS. 


Hydrocarbon. 

Heat  of 
Combustion. 

Difference. 

Ethylene   CaH4 

Cal. 
333  4 

Cal. 

Propylene    CaHe 

492  7 

159.3 

Isobutylene,  C4Hs            

650  6 

157.9 

Amylene,  CsHio        <  

807  6 

157.0 

Acetylene,  C2Ha  

310.1 

Allylene  C3F4 

467  6 

157.5 

*  Thermochemische  Untersuchungen,  4  Vols. 

f  Essai  de  Mecanique  Chimique,  Thermochimie,  Donees  et  Lois  Numeri- 
ques. 

$  Jour,  prakt.  Chem.,  1885  to  1895. 


THERMOCHEMISTRY  253 

ALCOHOLS. 


Alcohol. 

Heat  of 
Combustion. 

Difference. 

Methyl  alcohol   CH4O 

Cal. 

182  2 

Cal. 

Ethyl  alcohol,  C2H6O 

340  5 

158.3 

Propyl  alcohol,  C3H8O                                                . 

498.6 

158.1 

Isobutyl  alcohol,  C4Hi0O  

658.5 

159.9 

It  will  be  observed  that  a  very  nearly  constant  difference  in 
the  heat  of  combustion  corresponds  to  a  constant  difference  of  a 
CH2  group  in  composition.  A  number  of  interesting  relations 
between  heats  of  combustion  of  compounds  and  their  differences 
in  composition  have  been  discovered,  but  these  cannot  be  taken  up 
at  this  time.  It  has  also  been  pointed  out  that  the  heat  of  com- 
bustion of  organic  compounds  is  conditioned  not  only  by  their 
composition,  but  also  by  their  molecular  constitution. 

Some  exceedingly  interesting  and  important  results  have  been 
obtained  with  the  different  allotropic  forms  of  the  elements.  For 
example,  when  equal  masses  of  the  three  common  allotropic  forms 
of  carbon  are  burned  in  oxygen,  the  amounts  of  heat  evolved  are 
found  to  be  quite  different,  as  is  shown  by  the  following  energy 
equations :  — 

[C]  diamond  +  2  (0)  =  (C02)  +  94.3  Cal. 
[C]  graphite  +  2  (0)  =  (CO2)  +  94.8  Cal. 
[C]  amorphous  +  2  (0)  =  (C02)  +  97.65  Cal. 

It  is  apparent  that  amorphous  carbon  contains  the  greatest 
amount  of  energy  of  any  one  of  the  three  allotropic  modifications, 
and,  therefore,  when  amorphous  carbon  is  changed  into  diamond, 
the  reaction  must  be  accompanied  by  the  evolution  of  (97.65  — 
94.3)  =  3.35  Cal.  In  like  manner,  the  allotropic  forms  of  sulphur 
and  phosphorus  have  different  heats  of  combustion.  The  follow- 
ing equations  show  the  heat  equivalents  of  the  differences  in 
intrinsic  energy  between  the  allotropic  forms :  — 

S  (monoclinic)  =  S  (rhombic)  +  2.3  Cal. 
P  (white)  =  P  (red)  +  3.71  Cal. 


254  THEORETICAL  CHEMISTRY 

When  the  same  substance  is  burned  in  oxygen  and  then  in  ozone, 
it  is  found  that  more  heat  is  evolved  in  ozone  than  in  oxygen. 
The  energy  equation  expressing  the  change  of  ozone  into  oxygen 
may  be  written  thus, 

(08)  =  1}  (02)  +  36.2  Cal. 

All  of  the  above  facts  illustrate  the  general  principle  that  the 
larger  amounts  of  intrinsic  energy  are  associated  with  the  more 
unstable  forms. 

Thermoneutrality  of  Salt  Solutions.  In  addition  to  the  law 
of  constant  heat  summation,  Hess  discovered  two  other  important 
laws  of  thermochemistry,  viz.,  the  law  of  thermoneutrality  of 
salt  solutions,  and  the  law  governing  the  neutralization  of  acids 
by  bases.*  When  two  dilute  salt  solutions  are  mixed  there  is 
neither  evolution  nor  absorption  of  heat.  Thus  when  dilute  solu- 
tions of  sodium  nitrate  and  potassium  chloride  are  mixed,  there  is 
no  thermal  effect.  The  energy  equation  may  be  written  as  follows : — 

NaN03,  aq.  +  KC1,  aq.  =  NaCl,  aq.  +  KN03,  aq.  +  0  Cal. 

According  to  this  equation  a  double  decomposition  has  taken 
place  and  we  should  naturally  expect  an  evolution  or  an  absorption 
of  heat.  While  Hess  could  not  account  for  the  absence  of  any 
thermal  effect,  he  recognized  the  fact  as  quite  general  and  formu- 
lated the  law  of  the  thermoneutrality  of  salt  solutions  as  fol- 
lows:—  The  metathesis,  of  neutral  salts  in  dilute  solutions  takes  place 
with  neither  evolution  nor  absorption  of  heat. 

The  explanation  of  the  phenomenon  of  thermoneutrality  was 
furnished  by  the  theory  of  electrolytic  dissociation.  When  the 
above  equation  is  written  in  the  ionic  form,  it  becomes 

Na-  +  NO,'  +  1C  +  Cl'  =  Na-  +  Cl'  +  K'  +  NO/. 

From  this  it  is  apparent  that  the  same  ions  exist  on  both  sides  of 
the  equation,  and  in  reality  no  reaction  takes  place. 

There  are   numerous   exceptions   to  the   law  of   thermoneu- 
trality.    These  can  be  satisfactorily  accounted  for  by  the  theory  of 
electrolytic  dissociation.     All  of  those  salts  the  behavior  of  which  in 
dilute  solution  is  contrary  to  the  law,  are  found  to  be  only  partially 
*  Pogg.  Ann.,  50,  385  (1840). 


THERMOCHEMISTRY  255 

ionized,  and,  therefore,  when  their  solutions  are  mixed,  a  chem- 
ical reaction  actually  occurs.  The  exceptions  must  be  considered 
as  furnishing  additional  evidence  in  favor  of  the  theory  of  elec- 
trolytic dissociation. 

Heat  of  Neutralization.  Hess  also  discovered  *  that  when 
dilute  solutions  of  equivalent  quantities  of  strong  acids  and 
strong  bases  are  mixed,  practically  the  same  amount  of  heat  is 
evolved.  The  following  energy  equations  may  be  considered  as 
typical  examples  of  such  neutralizations :  — 

HC1,  aq.  +  NaOH,  aq.  =  NaCl,  aq.  +  H20  +  13.75  Cal., 
HN03,  aq.  +  NaOH,  aq.  =  NaN03  aq.  +  H20  +  13.68  Cal., 
HC1,  aq.  +  KOH,  aq.  =  KC1,  aq.  +  H20  +  13.70  Cal., 
HN03,  aq.  +  KOH,  aq.  =  KNO3,  aq.  +  H2O  +  13.77  Cal., 
HC1,  aq.  +  LiOH,  aq.  =  LiCl,  aq.  +  H20  +  13.70  Cal. 

Here  again  it  would  be  difficult  to  explain  the  phenomenon  with- 
out the  theory  of  electrolytic  dissociation.  In  terms  of  this 
theory,  however,  the  explanation  is  perfectly  plausible.  If  MOH 
and  HA  represent  any  strong  base  and  any  strong  acid  respectively, 
then  when  equivalent  amounts  of  these  are  dissolved  in  water, 
each  solution  being  largely  diluted  to  the  same  volume,  the  reac- 
tion may  be  written  thus :  — 

M-  +  OH'  +  H-  +  A'  =  M-  +  A'  +  H20  +  13.7  Cal. 

Disregarding  the  ions  which  occur  on  both  sides  of  the  equality 
sign,  we  have 

OH'  +  H-  =  H20  +  13.7  Cal. 

It  thus  appears  that  the  neutralization  of  a  strong  acid  by  a  strong 
base  in  dilute  solution  consists  solely  in  the  combination  of  hydro- 
gen and  hydroxyl  ions  to  form  undissociated  water,  the  heat  of 
this  ionic  reaction  being  13.7  large  calories. 

The  heat  of  formation  of  water  from  its  ions  must  not  be  con- 
fused with  the  heat  of  formation  of  water  from  its  elements. 
When  weak  acids  or  weak  bases  are  neutralized  by  strong  bases 
or  strong  acids,  or  when  weak  acids  are  neutralized  by  weak  bases, 

*  Loc.  cit. 


256  THEORETICAL  CHEMISTRY 

the  heat  of  neutralization  may  differ  widely  from  13.7  Cal.  This 
is  shown  by  the  following  thermochemical  equations :  — 

H-COOH,  aq.  +  NaOH,  aq.  =  H-COONa,  aq.  +  H20 

+  13.40  Cal., 
HC12-COOH,  aq.  +  NaOH,  aq.  =  HCl2-COONa,  aq.  +  H20 

+  14.83  Cal., 
H-COOH,  aq.  +  NH4OH,  aq.  =  H.COONH4,  aq.  +  H2O 

+  11.90  Cal., 
HCN,  aq.  +  NaOH,  aq.  =  NaCN,  aq.  +  H20  +  2.90  Cal. 

As  will  be  seen,  the  heat  of  neutralization  may  be  either  greater 
or  less  than  13.7  Cal.  The  exceptions  to  the  generalization  of 
constant  heat  of  neutralization  are  readily  explained  by  the 
theory  of  electrolytic  dissociation.  Suppose  a  weak  acid  to  be 
neutralized  by  a  strong  base.  According  to  the  dissociation 
theory,  the  acid  is  only  slightly  dissociated  and,  therefore,  yields 
a  comparatively  small  number  of  hydrogen  ions  to  the  solution. 
The  base  on  the  other  hand  is  completely  dissociated  into  hydroxyl 
and  metallic  ions.  Therefore,  as  many  hydroxyl  ions  disappear 
as  there  are  free  hydrogen  ions  with  which  they  can  combine  to 
form  water.  When  the  equilibrium  between  the  acid  and  the 
products  of  its  dissociation  has  been  thus  disturbed,  it  undergoes 
further  dissociation  and  the  resulting  hydrogen  ions  immediately 
combine  with  the  free  hydroxyl  ions  of  the  base.  This  process 
continues  until  all  of  the  hydroxyl  ions  of  the  base  have  been 
neutralized.  It  is  evident  that  the  thermal  effect  in  this  case  is 
the  algebraic  sum  of  the  heat  of  dissociation  of  the  weak  acid, 
which  may  be  positive  or  negative,  and  the  heat  of  formation  of 
water  from  its  ions.  A  similar  explanation  holds  for  the  neutrali- 
zation of  a  weak  base  by  a  strong  acid,  or  for  the  neutralization 
of  a  weak  acid  by  a  weak  base.  This  affords  a  method  for  estimat- 
ing the  approximate  value  of  the  heat  of  dissociation  of  a  weak 
acid  or  a  weak  base.  For  example,  in  the  equation  given  above, 

HCN,  aq.  +  NaOH,  aq.  =  NaCN,  aq.  +  H20  +  2.90  Cal., 

the  difference  between  2.9  and  13.7  or  —10.8  Cal.  represents 
approximately  the  heat  of  dissociation  of  hydrocyanic  acid. 


THERMOCHEMISTRY 


257 


Since  the  acid  is  initially  slightly  dissociated  in  dilute  solution, 
it  is  apparent  that  in  order  to  obtain  the  true  heat  of  dissociation, 
we  must  add  to  — 10.8  Cal.  the  thermal  value  of  the  dissociation 
of  that  portion  of  the  acid  which  has  already  become  ionized. 
Heat  of  lonization.  Since  13.7  Cal.  is  the  heat  of  formation 
of  water  from  its  ions,  this  must  also  be  the  thermal  equivalent  of 
the  energy  required  to  dissociate  one  mol  of  water  into  its  ions.  It 
must  be  remembered  that  the  dissociated  molecule  of  water  must 
bejnixed  with  a  very  large  volume  of  undissociated  water,  in  order 


HEAT  OF  FORMATION  OF  IONS. 


Ion. 

Heat  of 
Formation. 

Ion. 

Heat  of 
Formation. 

Hydrogen 

0.0 

Copper  (ic) 

—  15  8 

Potassium 

61.9 

Copper  (ous) 

—  16  0 

Sodium 

57.5 

Mercury  (ous)  .  .  . 

-19  8 

Lithium 

62.9 

Silver  

—25  3 

Ammonium                  .    . 

32.8 

Lead  

0  5 

Magnesium  . 

109.0 

Tin  (ous)  

3  3 

Calcium  ...             

109.0 

Chlorine  

39  3 

Aluminium     

121.0 

Bromine  

28.2 

Manganese  

50.2 

Iodine  

13  1 

Iron  (ous)  

22.2 

Sulphate  

214.4 

Iron  (ic)  

-9.3 

Sulphite  

151.3 

Cobalt. 

17  0 

Nitrous 

27  0 

Nickel 

16  0 

Nitric  . 

49  0 

Zinc.. 

35.1 

Carbonate 

161  1 

Cadmium 

18.4 

Hyroxyl 

54  7 

that  the  dissociation  may  be  permanent.  Reference  to  the  table 
of  heats  of  formation  (p.  246),  will  show  that  68.4  Cal.  are  required 
to  form  one  mol  of  water  from  its  elements.  Hence,  it  follows  that 
68.4  —  13.7  =  54.7  Cal.,  is  the  heat  of  formation  of  one  equivalent 
of  hydrogen  and  hydroxyl  ions.  It  has  been  shown  that  an  ex- 
tremely small  amount  of  energy  is  necessary  to  ionize  hydrogen 
when  it  is  dissolved  in  water.  It  is  evident,  therefore,  that  54.7 
Cal.  is  a  close  approximation  to  the  heat  of  formation  of  one  equiva- 
lent of  hydroxyl  ions. 

On  the  assumption  that  the  heat  of  ionization  of  gaseous  hydro- 
gen in  solution  is  zero,  the  values  of  the  other  ionic  heats  of  forma- 


258  THEORETICAL  CHEMISTRY 

tion  may  be  computed.  For  example,  the  heat  of  formation  of 
KOH,  aq.  is  116.5  Cal.  The  ionic  heat  of  formation  of  sodium 
ions  must  be  116.5  —  54.7  =  61.8  Cal.  In  like  manner,  the 
heat  of  formation  of  KC1,  aq.  is  101.2  Cal.;  hence  the  ionic  heat 
of  formation  of  chlorine  ions  must  be  101.2  —  61.8  =  39.4  Cal. 
The  preceding  table  of  heats  of  ionic  formation  has  been  calculated 
as  in  the  above  examples. 

The  Principle  of  Maximum  Work.  A  fundamental  principle 
of  the  science  of  mechanics  is  that  a  system  is  in  stable  equilib- 
rium when  its  potential  energy  is  a  minimum.  In  1879,  Ber- 
thelot  *  suggested  that  a  similar  principle  applies  to  chemical 
systems. 

In  terms  of  the  kinetic  theory,  the  temperature  of  a  substance 
is  to  be  regarded  as  a  measure  of  the  kinetic  energy  of  its  molecules. 
The  development  of  heat  by  a  chemical  reaction  would,  therefore, 
be  taken  as  an  indication  of  a  decrease  in  the  potential  energy  of 
the  system.  Berthelot's  theorem,  known  as  the  principle  of 
maximum  work,  may  be  stated  as  follows:  —  "Every  chemical  proc- 
ess accomplished  without  the  intervention  of  any  external  energy 
tends  to  produce  that  substance  or  system  of  substances  which  evolves 
the  maximum  amount  of  heat.}}  The  table  of  heats  of  formation 
(p.  246),  illustrates  the  general  truth  of  this  principle,  but  as  will 
be  seen,  the  theorem  precludes  the  possibility  of  spontaneous 
endothermic  reactions.  Thus,  for  example,  the  formation  of 
acetylene  from  its  elements  at  the  temperature  of  the  electric 
arc  is  a  well-known  endothermic  reaction,  but  according  to  the 
principle  of  maximum  work,  it  could  not  take  place  spontaneously. 
Another  serious  objection  to  Berthelot's  principle  is,  that  accord- 
ing to  it,  all  chemical  reactions  should  proceed  to  completion,  the 
reaction  taking  place  in  such  a  way  as  to  evolve  the  greatest  amount 
of  heat.  As  is  well  known,  many  reactions,  and  theoretically  all 
reactions,  are  never  complete,  but  proceed  until  a  condition  of 
equilibrium  is  reached.  The  principle  of  maximum  work,  there- 
fore, denies  the  existence  of  equilibria  in  chemical  reactions. 
Many  attempts  have  been  made  to  "explain  away"  these  defects, 
but  none  of  them  have  been  successful.  In  referring  to  the  generali- 
*  Essai  de  Mecanique  Chimique. 


THERMOCHEMISTRY  259 

zation,  Le  Chatelier  terms  it  "a  very  interesting  approximation 
toward  a  strictly  valid  generalization. " 

The  Theorem  of  Le  Chatelier.  As  a  result  of  his  attempts  to 
modify  the  principle  of  maximum  work  and  render  it  generally 
applicable,  Le  Chatelier  was  led  to  the  discovery  of  a  rigorous  law 
of  wide-reaching  usefulness.  His  generalization  may  be  stated 
as  follows:  —  Any  alteration  in  the  factors  which  determine  an  equi- 
librium, causes  the  equilibrium  to  become  displaced  in  such  a  way 
as  to  oppose,  as  far  as  possible,  the  effect  of  the  alteration.  If  the 
temperature  of  a  system  which  is  in  equilibrium  be  raised  or 
lowered,  the  resulting  displacement  of  the  equilibrium  is  accom- 
panied by  such  absorption  or  evolution  of  heat  as  will  tend  to 
maintain  the  temperature  constant.  An  interesting  illustration 
of  the  behavior  of  a  system  when  one  of  the  factors  controlling 
the  equilibrium  is  varied,  is  afforded  by  the  system 

2N02<=±N204. 

The  reaction  proceeds  in  the  direction  indicated  by  the  upper 
arrow  with  the  evolution  of  12.6  Cal.  Increase  of  temperature 
favors  the  reaction  which  is  accompanied  by  an  absorption  of 
heat,  which  in  this  case,  is  the  reaction  indicated  by  the  lower 
arrow.  Hence  as  the  temperature  rises,  the  percentage  of  NO2 
increases  at  the  expense  of  N2O4.  This  fact  can  be  demonstrated 
by  the  following  experiment.  Some  liquefied  N2O4  is  placed  in 
each  of  three  long  glass  tubes,  which  are  sealed  at  one  end.  When 
enough  N204  has  vaporized  to  displace  the  air,  the  open  ends  of 
the  tubes  are  sealed.  Changes  in  the  equilibrium  caused  by 
varying  the  temperature  can  be  followed  by  noting  the  changes 
in  the  color  of  the  mixture.  N204  is  an  almost  colorless  substance, 
while  N02  is  reddish  brown.  At  ordinary  temperatures  the 
contents  of  the  tubes  will  be  brown  in  color.  One  tube  is  set 
aside  as  a  standard  of  comparison,  while  the  temperature  of  the 
second  is  lowered  by  surrounding  it  with  a  freezing  mixture.  As 
the  temperature  falls,  the  brown  color  of  the  contents  of  the  tube 
becomes  much  lighter,  showing  an  increased  formation  of  N2C>4. 
The  third  tube  is  heated  by  immersing  it  in  a  beaker  of  boiling 
water.  As  the  temperature  rises,  the  contents  of  the  tube  becomes 


260  THEORETICAL  CHEMISTRY 

much  darker  in  color,  indicating  an  increase  in  the  amount  of  NC>2 
in  the  mixture. 

Another  example  is  afforded  by  the  equilibrium  between  ozone 
and  oxygen,  represented  by  the  equation 


The  reaction  indicated  by  the  upper  arrow  is  exothermic.  In- 
crease of  temperature  causes  a  displacement  of  the  equilibrium 
in  the  direction  of  the  lower  arrow,  since  under  these  conditions 
heat  is  absorbed.  Thus,  as  the  temperature  rises  ozone  becomes 
increasingly  stable.  Nernst  has  calculated  that  at  6000°  C., 
the  temperature  of  the  photosphere  of  the  sun,  10  per  cent  of  the 
above  equilibrium  mixture  would  be  ozone.  Other  applications 
of  the  theorem  of  Le  Chatelier  will  be  given  in  subsequent 
chapters. 

PROBLEMS. 

1.  From  the  following  data  calculate  the  heat  of  formation  of  HN02 
aq.— 

[NH4N02]  =  (N2)  +  2  H20  +  71.77  Gal, 

2  (H2)  +  (Oi)  =  2  H20  +  136.72  Cal., 

(N2)  +  3  (H2)  +  aq.  =  2  NH3  aq.  +  40.64  Cal., 

NH3aq.  +  HN02aq.  =  NH4N02aq.  +  9.110  Cal., 

[NH4N02]  +  aq.  =  NH4N02aq.  -  4.75  Cal. 

Am.   (H)  +  (N)  +  (02)  +  aq.  =  HN02  aq.  +  30.77  Cal. 

2.  By  the  combustion  at  constant  pressure  of  2  grams  of  hydrogen 
with  oxygen  to  form  liquid  water  at  17°  C.,  68.36  cal.  are  evolved.     What 
is  the  heat  evolution  at  constant  volume?  Ans.   67.49  Cal. 

3.  The  heats  of  solution  of  Na2S04,  Na2S04.H20,  and  Na2S04.10H20 
are  0.46,   —  1.9  and  —  18.76  Cal.  respectively.    What  are  the  heats  of 
hydration  of  Na2S04;   (a)  to  monohydrate,  (b)  to  decahydrate? 

Ans.   (a)  2.36  Cal.,  (b)  19.22  Cal. 

£     4.   The  heats  of  neutralization  of  NaOH  and  NH4OH  by  HC1  are  13.68 

(  and  12.27  Cal.  respectively.    What  is  the  heat  of  ionization  of  NH4OH, 

if  it  is  assumed  to  be  practically  undissociated?  Ans.  1.41  Cal. 


THERMOCHEMISTRY  261 

5.  From  the  following  energy  equations  :  — 
[C]  +  (02)  =  (COO  +  96.96  Cal., 
2  (HO  +  (02)  =  2  H20  +  136.72  Cal., 
2  C6H6  +  15  (00  =  12  (C02)  +  6  H20  +  1598.7  Cal., 
2  (C2HO  +  5  (00  =  4  (COO  +  2  H20  +  620.1  Cal., 

all  at  17°  C.  and  constant  pressure,  calculate  the  heat  evolved  at  17°  C. 
in  the  reaction 

3  C2H2  = 


(a)  at  constant  pressure,  and  (b)  at  constant  volume. 

Ans.   (a)  130.8  Cal.,  (b)  129.64  cal. 

6.  Calculate  the  heat  of  formation  of  sulphur  trioxide  from  the  follow- 
ing energy  equations  :  — 

[PbO]  +  [S]  +  3  (0)  =  [PbS04]  +  1655  K. 
[PbO]  +  H2S04.5  H20  =  [PbS04]  +  6  H20  +  233  K. 
[S]  +  3  (0)  +  6  H20  =  H2S04.5  H20  +  1422  K. 
[SOa]  +  6  H20  =  H2S04.5  H20  +  411  K. 

Ans.  [S]  +  3  (0)  =  [80s]  +  1011  K. 

7.  What  is  the  heat  of  formation  of  a  very  dilute  solution  of  calcium 
chloride?     (See  table  on  p.  257.)  Ans.   187.6  Cal. 


CHAPTER   XII. 

HOMOGENEOUS   EQUILIBRIUM. 

Historical  Introduction.  In  this  and  the  two  succeeding 
chapters  the  conditions  which  affect  the  rate  and  the  extent  of 
chemical  reactions  will  be  considered.  When  two  substances 
react  chemically,  it  is  customary  to  refer  the  phenomenon  to  the 
existence  of  an  attractive  force  known  as  chemical  affinity. 

Ever  since  the  metaphysical  speculations  of  the  Greeks,  who 
endowed  the  atoms  with  the  instincts  of  love  and  hate,  the  nature 
of  chemical  affinity  has  been  under  discussion.  So  little  has  been 
learned  as  to  the  cause  of  chemical  reactions,  that  in  recent  years 
this  question  has  been  dismissed  and  attention  has  been  directed 
to  the  more  promising  question  as  to  how  they  take  place.  New- 
ton's discovery  of  the  law  of  gravitation  led  him  to  consider  the 
attraction  between  atoms  and  the  attraction  between  large  masses 
of  matter  as  manifestations  of  the  same  force. 

Although  Newton  found  that  chemical  attraction  does  not 
follow  the  law  of  the  inverse  square,  yet  his  suggestion  exerted 
a  profound  influence  upon  the  minds  of  his  contemporaries. 

Geoffroy  and  Bergmann  arranged  chemical  substances  in  the 
order  of  their  displacing  power.  Thus,  if  we  have  three  sub- 
stances, A,  B,  and  C  and  the  attraction  between  A  and  B  is 
greater  than  that  between  A  and  (7,  then  when  B  is  added  to  AC 
it  will  completely  displace  C,  as  indicated  by  the  following  equa- 
tion:- AC  +  B  =  AB  +  C. 

These  investigators  overlooked  a  factor  of  fundamental  importance 
in  conditioning  chemical  reactivity,  viz.,  the  influence  of  mass. 
The  importance  of  the  relative  amounts  of  the  reacting  substances 
in  determining  the  course  of  a  reaction  was  first  clearly  recognized 
by  Wenzel  *  in  1777.  It  remained  for  Berthollet,f  however,  to 

*  Lehre  von  der  chemischen  Verwandtschaft  der  Korper. 
t  Essai  de  Statique  Chimique. 
262 


HOMOGENEOUS  EQUILIBRIUM  263 

point  out  the  significance  of  the  views  advanced  by  Wenzel. 
His  first  paper  on  this  subject  was  published  in  1799,  while  acting 
as  a  scientific  adviser  to  Napoleon  on  his  Egyptian  expedition. 
Under  ordinary  conditions  sodium  carbonate  and  calcium  chloride 
react  according  to  the  equation, 

Na2C03  +  CaCl2  =  2  NaCl  +  CaC03, 

the  reaction  proceeding  nearly  to  completion.  Berthollet  observed 
the  deposits  of  sodium  carbonate  on  the  shores  of  certain  saline 
lakes  in  Egypt,  and  pointed  out  that  this  salt  is  produced  by  the 
reversal  of  the  above  reaction,  the  large  excess  of  sodium  chloride 
in  solution  in  the  water  of  the  lakes  conditioning  the  course  of 
the  reaction. 

The  German  chemist  Rose*  furnished  much  additional  evidence 
in  favor  of  the  effect  of  mass  on  chemical  reactions.  He  pointed 
out  that  in  nature,  the  silicates,  which  are  among  the  most  stable 
compounds  known,  are  undergoing  a  continual  decomposition 
under  the  influence  of  such  relatively  weak  agents  as  water  and 
carbon  dioxide.  The  relatively  strong  specific  affinities  of  the 
atoms  of  the  silicates  are  overcome  by  the  preponderating  masses 
of  water  and  carbon  dioxide  in  the  atmosphere.  In  1862  an 
important  contribution  to  our  knowledge  of  the  effect  of  mass  on 
the  course  of  a  chemical  reaction  was  made  by  Berthelot  and  Pean 
de  St.  Gilles.f  They  investigated  the  formation  of  esters  from 
alcohols  and  acids.  The  reaction  between  ethyl  alcohol  and 
acetic  acid  is  represented  by  the  equation 

C2H5OH  +  CHsCOOH  +±  CH3COOC2H5  +  H20. 

Starting  with  equivalent  quantities  of  alcohol  and  acid,  the  reac- 
tion proceeds  until  about  two-thirds  of  the  reacting  substances 
have  been  converted  into  ester  and  water.  In  like  manner,  if 
equivalent  quantities  of  ethyl  acetate  and  water  are  brought 
together,  the  reaction  proceeds  in  the  direction  indicated  by  the 
lower  arrow,  until  about  one-third  of  the  original  substances  have 
been  converted  into  acid  and  alcohol.  In  other  words  the  reac- 
tion is  reversible,  a  condition  of  equilibrium  resulting  when  the 

*  Pogg.  Ann.,  94,  481  (1855);  95,  96,  284,  426  (1855). 

t  Ann.  Chim.  Phys.  [3],  65,  385;  66,  5;  68,  225  (1862-1863), 


264 


THEORETICAL  CHEMISTRY 


speeds  of  the  two  reactions,  indicated  by  the  upper  and  lower 
arrows,  become  equal.  If  now  a  fixed  amount  of  acid  is  taken, 
say  1  equivalent,  and  the  quantity  of  alcohol  is  varied,  a  corre- 
sponding displacement  of  the  equilibrium  follows. 

The  following  table  gives  the  results  obtained  by  Berthelot  and 
Pean  de  St.  Gilles  for  ethyl  alcohol  and  acetic  acid.  The  first 
and  third  columns  give  the  number  of  equivalents  of  alcohol  to 
1  equivalent  of  acetic  acid,  and  the  second  and  fourth  columns 
give  the  percentage  of  ester  formed. 


Equivalents 
of  Alcohol. 

Ester 
Formed. 

Equivalents 
of  Alcohol. 

Ester 
Formed. 

0.2 

19.3 

2.0 

82.8 

0.5 

42.0 

4.0 

88.2 

1.0 

66.5 

12.0 

93.2 

1.5 

77.9 

50.0 

100.0 

The  effect  of  increasing  the  mass  of  alcohol  on  the  course  of  the 
reaction  is  very  beautifully  shown  by  the  above  results. 

The  Law  of  Mass  Action.  While  the  influence  of  the  relative 
masses  of  the  reacting  substances  in  conditioning  chemical  reac- 
tions was  thus  fully  established,  it  was  not  until  1867  that  the  law 
governing  the  action  of  mass  was  accurately  formulated. 

In  that  year  Guldberg  and  Waage,*  two  Scandinavian  investiga- 
tors, enunciated  the  law  of  mass  action  as  follows: —  The  amount 
of  chemical  action  at  any  stage  of  a  reaction  is  proportional  to  the 
active  masses  of  the  reacting  substances  present  at  that  time.  Guld- 
berg and  Waage  defined  the  term  " active  mass"  as  the  molecular 
concentration  of  the  reacting  substances.  It  is  to  be  carefully 
noted  that  the  amount  of  chemical  action  is  not  porportional 
to  the  actual  masses  of  the  substances  present,  but  rather  to  the 
amounts  present  in  unit  volume.  The  law  is  generally  applicable 
to  homogeneous  systems;  that  is,  to  those  systems  in  which 
ordinary  observation  fails  to  reveal  the  presence  of  essentially 
different  parts.  The  amount  of  chemical  action  exerted  by  a 

*  Etudes  sur  les  Affinite's  Chimiques,  Jour,  prakt.  Chem.  [2],  19,  69  (1879). 


HOMOGENEOUS  EQUILIBRIUM  265 

substance  can  be  determined,  either  from  its  effect  on  the  equili- 
brium, or  from  its  influence  on  the  speed  of  reaction. 

In  order  to  apply  the  law  of  mass  action  practically,  it  must  be 
formulated  mathematically.  Let  a  and  b  denote  the  molecular 
concentrations  of  the  substances  initially  present  in  a  reversible 
reaction.  According  to  the  law  of  mass  action,  the  rate  at  which 
these  substances  combine  is  proportional  to  the  active  masses 
of  each  constituent,  and  therefore  to  their  product,  ab.  The 
initial  speed  of  the  reaction  at  the  time  £o  is  therefore, 

Speed<0  oo  ab,  or  Speedy  =  k  •  ab, 

in  which  the  proportionality  factor  k,  is  known  as  the  velocity 
constant.  As  the  reaction  proceeds,  the  molecular  concentrations 
of  the  original  substances  steadily  diminish,  while  the  molecular 
concentrations  of  the  products  of  the  reaction  steadily  increase. 
Let  us  assume  that  after  the  interval  of  time  t,  x  equivalents  of 
the  products  of  the  reaction  have  been  formed.  The  speed  of 
the  original  reaction  will  now  be 

Speedf  =  k  (a  —  x)  (b  —  x). 

As  the  reaction  proceeds  the  tendency  of  the  products  to  combine 
and  reform  the  original  substances  increases.  At  the  time  t, 
when  the  concentration  of  the  products  is  x,  the  speed  of  the 
reverse  reaction  will  be 

Speed*  =  ki  •  #2, 

where  ki  is  the  velocity  constant  of  the  reverse  reaction. 

We  thus  have  two  reactions  proceeding  in  opposite  directions: 
the  speed  of  the  direct  reaction  continuously  diminishes  while 
that  of  the  reverse  reaction  continually  increases.  It  is  evident 
that  a  point  must  ultimately  be  reached  at  which  the  speeds  of 
the  direct  and  reverse  reactions  become  equal,  and  a  condition 
of  equilibrium  will  be  established.  Let  Xi  represent  the  value  of 
x  when  equilibrium  is  attained;  we  then  have 

Speed  direct  =  k  (a  -  Xi)  (b  -  Zi)  =  Speed  reverse 

or 

(a  -  Xi)  (b  -  xi)  _  &i  _ 

~ 


266  THEORETICAL  CHEMISTRY 

in  which  K  is  known  as  the  equilibrium  constant.  Since  the  veloc- 
ity constants  k  and  k\,  are  independent  of  the  concentration,  it 
follows  that  the  above  equation  holds  for  all  concentrations. 
Therefore,  if  the  value  of  the  equilibrium  constant  of  a  reaction 
is  known,  the  equilibrium  conditions  can  be  calculated  for  any 
concentrations  of  the  reacting  substances.  When  more  than  one 
mol  of  a  substance  is  involved  in  a  reaction,  each  mol  must  be 
considered  separately  in  the  mass  action  equation. 
Thus,  let 

niAi  +  n2A2  +  .  .  .  +±  ni'Ai  +  n2'A2'  +  .  .  . 

represent  any  reversible  reaction,  in  which  n\  mols  of  A\  and  n2 
mols  of  A2  react  to  form  n\  mols  of  A\  and  n2  mols  of  A2.  When 
equilibrium  is  attained,  we  shall  have 


or 


in  which  the  symbol  c  is  used  to  denote  the  active  mass,  or  molecular 
concentration  of  the  substances  involved  in  the  reaction.  This 
.is  a  perfectly  general  form  of  the  mass-action  equation.  Since  at 
any  one  temperature,  concentration  and  pressure  are  proportional, 
we  may  write  equation  (1)  in  the  following  form 


which,  in  the  case  of  gaseous  equilibria,  is  often  a  more  convenient 
form  of  the  equation. 
The  relation  between  the  two  equilibrium  constants,  Kc  and  Kp, 

can  be  easily  determined,  as  follows:  —  Since  c  =  -  =  -—,   we 

V        Kl 

have,  on  substituting  this  value  of  c  in  equation  (1), 

'jLW-aY* 

&T   \RT)   •  *  • 


BT      RT 


HOMOGENEOUS  EQUILIBRIUM  267 

or  indicating  the  sum  of  the  initial  number  of  mols  by  Sn  and  the 
sum  of  the  final  number  of  mols  by  2n',  we  have 
KC  =  KP(RT)  **-**. 

It  is  evident,  therefore,  that  in  reactions  where  the  same  number 
of  mols  occur  on  both  sides  of  the  equality  sign,  Kc  =  Kp.  Equa- 
tion (1)  (or  equation  (2))  is  sometimes  known  as  the  reaction 
isotherm.  While  the  law  of  mass  action  may  be  proved  thermody- 
namically,  a  much  simpler  kinetic  derivation  has  been  given  by 
Van't  Hoff.  If  we  assume  that  the  rate  of  chemical  change  is  pro- 
portional to  the  number  of  collisions  per  unit  of  time  between  the 
molecules  of  the  reacting  substances,  then  in  the  reaction 


the  velocity  of  the  direct  change  will  be  kc^c^  .  .  .  and  the 
velocity  of  the  reverse  reaction  will  be  kic^c^,.  •  •  • 

At  equilibrium,  the  two  velocities  will  be  equal  and,  therefore, 


/ccn;c? 


or 


Wi        719  Z* 

c    c 

As  a  consequence  of  the  assumptions  involved  in  both  the  thermo- 
dynamic  and  the  kinetic  proofs  of  the  law  of  mass  action,  it  fol- 
lows that  the  law  is  only  strictly  applicable  to  very  dilute  solutions. 
Notwithstanding  this  limitation,  experimental  results  indicate 
that  it  frequently  holds  for  moderately-concentrated  solutions. 

Equilibrium  in  Homogeneous  Gaseous  Systems. 

(a)  Decomposition  of  Hydriodic  Acid.  A  typical  example  of 
equilibrium  in  a  gaseous  system  is  afforded  by  the  decomposition 
of  hydriodic  acid,  as  represented  by  the  equation 


This  reaction  has  been  thoroughly  investigated  by  Hautefeuille, 
Lemoine  and  Bodenstein.*  The  reaction  is  well  adapted  for 
investigation  since  it  proceeds  very  slowly  at  ordinary  temper- 

*  Zeit.  Phys.  Chem.  22,  1  (1897). 


268  THEORETICAL  CHEMISTRY 

atures,  while  at  the  temperature  of  boiling  sulphur,  448°  C., 
equilibrium  is  established  quite  rapidly.  If  the  mixture  of.  gases 
is  maintained  at  448°  C.  for  some  time  and  is  then  cooled  quickly, 
the  respective  concentrations  of  the  components  of  the  mixture 
can  be  determined  by  the  ordinary  methods  of  chemical  analysis. 
Various  mixtures  of  the  gases  are  sealed  in  glass  tubes  and  heated 
for  a  definite  time  in  the  vapor  of  boiling  sulphur.  The  tubes 
are  then  cooled  rapidly  to  the  temperature  of  the  room  and,  after 
the  iodine  and  hydriodic  acid  have  been  removed  by  absorption  in 
potassium  hydroxide,  the  amount  of  free  hydrogen  present  in  each 
tube  is  measured. 

Applying  the  law  of  mass  action  to  the  above  equation,  we  have 


Expressing  the  analytical  results  in  mols,  let  a  mols  of  iodine  be 
mixed  with  b  mols  of  hydrogen,  and  let  2  x  mols  of  hydriodic  acid 
be  formed.  Then  when  equilibrium  is  established,  a  —  x  will  be 
the  amount  of  iodine  vapor  and  b  —  x  will  be  the  amount  of  hydro- 
gen present.  The  concentrations  being  directly  proportional  to 
the  amounts  present,  we  may  substitute  these  values  for  cHz,  c/2, 
and  CHI  in  the  mass-action  equation.  The  following  expression 
is  thus  obtained:  — 

(b  -x)(a-  x)  _ 
-~  Kc' 


Solving  the  equation  for  x,  we  obtain 


_  a  4.  &  -  Vq2  +  &2  _  ab  (2  -  16  j 


Since,  according  to  Avogadro's  law,  equal  volumes  of  all  gases 
contain  the  same  number  of  molecules,  volumes  may  be  sub- 
stituted for  a,  b,  and  x.  Bodenstein  expressed  his  results  in  terms 
of  volumes  reduced  to  standard  conditions  of  temperature  and 
pressure.  On  analyzing  equilibrium  mixtures,  Bodenstein  found 
that  at  448°  C.,  Kc  =  0.01984,  and  at  350°  C.,  Kc  =  0.01494. 

Having  determined  the  value  of  the  equilibrium  constant,  he 
made  use  of  this  value  in  calculating  the  volume  of  hydriodic 


HOMOGENEOUS  EQUILIBRIUM 


269 


acid  which  should  be  obtained  from  known  volumes  of  hydrogen 
and  iodine.  A  comparison  of  the  calculated  and  observed  values 
showed  excellent  agreement.  The  following  table  contains  a  few 
of  the  results  obtained  by  Bodenstein  at  448°  C. 


Hydrogen, 
b. 

Iodine, 
a. 

HI  calculated, 
2z. 

HI  observed, 
2z. 

20.57 

5.22 

10.19 

10.22 

20.60 

14.45 

25.54 

25.72 

15.75 

11.90 

20.65 

20.70 

14.47 

38.93 

27.77 

27.64 

8.10 

2.94 

5.64 

5.66 

8.07 

9.27 

13.47 

13.34 

It  is  of  interest  to  note  that  a  change  in  pressure  does  not 
alter  the  equilibrium  in  this  gaseous  system.  Making  use  of  the 
partial  pressures  of  the  components  of  the  gaseous  system  instead 
of  the  concentrations,  we  have 


2 
P 


HI 


Now  let  the  total  pressure  on  the  system  be  increased  to  n  times 
its  original  value;  then  the  partial  pressures  are  all  increased  in 
the  same  proportion,  and  we  have 


-2.7,2        •       P> 
71  PHI 

which  is  equivalent  to  the  original  expression,  since  n  cancels 
out.  The  equilibrium  is  thus  seen  to  be  independent  of  the  pres- 
sure. This  is  only  true  for  those  systems  in  which  a  change  in 
volume  does  not  occur. 

(b)  Dissociation  of  Phosphorus  Pentachloride.  When  phos- 
phorus pentachloride  is  vaporized  it  dissociates  according  to  the 
following  equation 


Applying  the  law  of  mass  action,  we  have 


270  THEORETICAL  CHEMISTRY 

Starting  with  1  mol  of  phosphorus  pentachloride,  which  if  undis- 
sociated  would  occupy  the  volume  7',  under  atmospheric  pressure, 
and  letting  a  denote  the  degree  of  dissociation,  the  molecular  con- 
centrations at  equilibrium  will  be  as  follows:  — 

~ 


=  (l+a)  CPCI>  =  (1+)7"  "         **  =  (\+a)V>' 

Letting  (1  +  a)  V  =  V,  and  substituting  in  the  above  equation, 
we  have 

-  X- 

F     -F  a2  K 

I-  a        (l-a)V  °' 


At  250°  C.  phosphorus  pentachloride  is  dissociated  to  the  extent 
of  80  per  cent.     Under  atmospheric  pressure  1  mol  will  be  present 


070    I 

in  22.4-    Jl-  liters  =  V  '.    The  final  volume  will,  therefore,  be 
2ti  o 


The  value  of  the  equilibrium  constant,  —  usually  designated  in 
cases  of  dissociation,  the  dissociation  constant,  —  is,  therefore, 

= (0.8)2 

(1  -  0.8)  ( 

Having  obtained  the  value  of  KC)  the  direction  and  extent  of  the 
reaction  at  250°  C.  can  be  determined,  provided  the  initial  molec- 
ular concentrations  are  known.  The  reaction  is  accompanied  by 
a  change  in  volume,  and,  therefore,  the  equilibrium  is  displaced  by 
a  change  in  pressure.  Making  use  of  the  partial  pressures  of  the 
components  of  the  gaseous  mixture,  we  have 

pi  =  Kp' 

where  pi  and  p%  are  the  partial  pressures  of  phosphorus  penta- 
chloride and  the  products  of  the  dissociation,  phosphorus  tri- 


HOMOGENEOUS  EQUILIBRIUM  271 

chloride  and  chlorine,  respectively.     Let  the  total  pressure  be 
increased  n-times,  then 


It  is  apparent  from  this  equation,  that  the  equilibrium  is  not 
independent  of  the  pressure,  an  increase  in  pressure  being  accom- 
panied by  a  diminution  of  the  dissociation.  An  important  point 
in  connection  with  dissociation,  first  observed  by  Deville,*  is  the 
effect  on  the  equilibrium  of  the  addition  of  an  excess  of  one  of  the 
products  of  dissociation.  For  example,  in  the  equilibrium 


an  excess  of  chlorine  or  of  phosphorus  trichloride,  drives  back  the 
dissociation.  If  p\  denotes  the  partial  pressure  of  phosphorus 
pentachloride,  p2  that  of  phosphorus  trichloride,  and  p$  that  of 
chlorine,  then  we  have 


Now  let  an  excess  of  chlorine  be  added;  this  will  cause  the  value 
of  ps  to  increase.  Since  the  value  of  Kp  is  constant,  the  value  of 
PZ  must  diminish  and  that  of  p\  must  increase.  Hence,  the  addi- 
tion of  an  excess  of  either  product  of  dissociation  causes  a  diminu- 
tion of  the  amount  of  the  dissociation. 

(c)  Dissociation  of  Carbon  Dioxide.     Carbon  dioxide  dissociates 
according  to  the  equation, 

2C02<=>2CO  +  02. 

This  is  a  somewhat  more  complex  gaseous  system  than  either  of 
the  foregoing  systems.  When  equilibrium  is  established,  let 
pi  be  the  partial  pressure  of  the  carbon  dioxide,  p2  the  partial  pres- 
sure of  carbon  monoxide,  and  ps  the  partial  pressure  of  oxygen, 
then  we  have 

P22  *  PS  _  K 

-  o  —    —    f^-'O* 
Pi2 

At  3000°  C.  and  under  atmospheric  pressure,  carbon  dioxide  is 
*  Lemons  sur  la  dissociation,  Paris  (1866). 


272  THEORETICAL  CHEMISTRY 

40  per  cent  dissociated.     The  partial  pressures  of  each  of  the  com- 
ponents may  be  readily  calculated  as  follows:  — 
=  2  (1  -  0.40) 

2  (1  -  0.40)  +  3  X  0.40 

2X0.40 
2  (1  -  0.40)  +  3  X  0.40 

0.40 
P*  =  2  (1  -  0.40)  +  3  X  0.40  =  °'17- 

Substituting  these  values  in  the  above  equation,  we  obtain 

(0.33)2  X  0.17 


'  (0.50)* 

The  dissociation  constant  for  carbon  dioxide  may  have  a  different 
value  if  the  equation  is  written  in  the  form 


Applying  the  law  of  mass  action,  we  have 


. 

Pi 

Substituting  the  above  values  of  the  partial  pressures,  we  obtain 

Kp  =  0.272. 

Equilibrium  in  Liquid  Systems.     The  reaction  between  an 
alcohol  and  an  acid  to  form  an  ester  and  water  may  be  taken  as 
an  example  of  equilibrium  in  a  liquid  system.     In  the  reaction 
C2H5OH  +  CHaCOOH^  CH3COOC2H6  +  H20, 

let  a,  b,  and  c  represent  the  number  of  mols  of  alcohol,  acid  and 
water  respectively,  which  are  present  in  V  liters  of  the  mixture, 
and  let  x  denote  the  number  of  mols  of  ester  and  water  which 
have  been  formed  when  the  system  has  reached  equilibrium. 
The  active  masses  of  the  components  will  then  be, 

a  ~  x  b  ~  x 


-  r       -  .    n        -.      j     n 

v^alc.  —       "y      >      ^acid  -         y       J      tester  —  T?  J  and      Cwater 

Applying  the  law  of  mass  action,  we  obtain 
(a  -  x)  (b  -  x)  _ 


HOMOGENEOUS  EQUILIBRIUM 


273 


In  this  case  the  value  of  the  equilibrium  constant  is  independent 
of  the  volume.  This  reaction  has  been  studied,  as  already  men- 
tioned, by  Berthelot  and  Pean  de  St.  Gilles.*  They  found  that 
when  equivalent  amounts  of  alcohol  and  acid  are  mixed,  the  reac- 
tion proceeds  until  two-thirds  of  the  mixture  is  changed  into  ester 
and  water.  Hence,  we  find 


Kc 


_ 
ixt 


|. 


Having  determined  the  value  of  KC)  it  may  now  be  used  to  cal- 
culate the  equilibrium  conditions  for  any  initial  concentrations 
of  the  substances  involved  in  the  reaction.  As  an  illustration, 
we  will  take  1  mol  of  acetic  acid  and  treat  it  with  varying  amounts 
of  alcohol,  the  initial  mixture  containing  neither  of  the'products  of 
the  reaction.  The  equation  takes  the  form 
(a  -  x)  (1  -  x)  _  1 


Solving  for  x,  we  have  . 

x  =  f  (1  +  a  -  Va?  -  a 


1). 


A  comparison  of  the  observed  and  calculated  values  given  in  the 
accompanying  table  shows  that  the  agreement  is  excellent,  even 
in  the  more  concentrated  solutions,  where  we  might  reasonably 
expect  that  the  mass  law  would  cease  to  hold. 


Alcohol, 
a. 

Ester 
(observed), 
x. 

Ester 
(calculated), 
x. 

0.05 

0.05 

0.049 

0.08 

0.078 

0.078 

0.18 

0.171 

0.171 

0.28 

0.226 

0.232 

0.33 

0.293 

0.311 

0.50 

0.414 

0.523 

0.67 

0.519 

0.528 

1.0 

0.665 

0.667 

1.5 

0.819 

0.785 

2.0 

0.858 

0.845 

2.24 

0.876 

0.864 

8.0 

0.966 

0.945 

*  LOG.  cit. 


274  THEORETICAL  CHEMISTRY 

The  Variation  of  the  Equilibrium  Constant  with  Temperature. 

Van't  Hoff  showed  that  the  displacement  of  equilibrium  due  to 
change  in  temperature  is  connected  with  the  heat  evolved  or  ab- 
sorbed in  a  chemical  reaction  by  the  equations 

d(\ogeKp)_    Qv 
dT  RT2' 

and 

d(\ogeKp)  _    Qp 

dT  RT*' 

where  Qv  and  Qp  are  the  heats  of  reaction  at  constant  volume  and 
constant  pressure  respectively,  and  where  R  and  T  have  their 
usual  significance.  Either  form  of  the  equation,  known  as  the 
reaction  isochore,  shows  that  the  rate  of  change  of  the  natural 
logarithm  of  the  equilibrium  constant  with  temperature  is  equal 
to  the  total  heat  of  reaction  divided  by  the  molecular  gas  constant 
times  the  square  of  the  absolute  temperature  at  which  the 
reaction  takes  place.  Equations  (1)  and  (2)  hold  only  for  displace- 
ments of  the  equilibrium  due  to  infinitely  small  changes  in  temper- 
ature. In  order  to  render  these  equations  applicable  to  concrete 
equilibria,  it  is  necessary  to  integrate  them.  The  integration  of 
these  expressions  can  only  be  performed  if  Q  is  constant.  For 
small  intervals  of  temperature,  Q  is  practically  independent  of 
the  temperature,  and  for  larger  intervals  we  may  take  the  value 
of  Q  which  corresponds  to  the  mean  of  the  two  temperatures 
between  which  the  integration  is  performed.  Integrating  equa- 
tions (1)  and  (2)  on  this  assumption,  we  obtain 


log,Kc,  -  lo&Kc,=    «        -    T,  (3) 

and 

log.  KPI  -  log.  Kp,  =         i  -  i   •  (4) 


Passing  to  Briggsian  logarithms,  and  putting  R  =  1.99  calories, 
equations  (3)  and  (4)  become 

(5) 


HOMOGENEOUS  EQUILIBRIUM  275 

and 

(6) 


We  shall  now  proceed  to  show  how  these  important  equations 
may  be  applied  to  several  typical  equilibria. 

(a)  Vaporization  of  Water.     The  equilibrium  between  a  liquid 
and   its  vapor   is   conditioned   by    the   pressure   of   the    vapor, 
this  in  turn  being  dependent  upon  the  temperature.     In  this  case 
of  physical  equilibrium,  we  have  KP1  =  pif  and  KP2  =  p2.     The 
value  of  Qp  for  water  can  be  calculated  from  the  following  data:  — 

TI  =  273°,  pi  =    4.54  mm.  of  mercury, 

T2  =  273°  +11°.54,  p2  =  10.02  mm.  of  mercury. 

Substituting  in  equation  (6),  we  have 

n    _  4.581  (log  10.02  -  log  4.54)  273  X  284.5       . 
^p~  273  -  284.5 

or  Qp  =  —  10,670  calories. 

The  value  of  Qp  obtained  by  experiment  is  —10,854  calories. 

(b)  Dissociation  of  Nitrogen  Tetroxide.     In  the  reaction 


the  following  values  for  the  dissociation  of  N204  have  been  ob- 

tained :  — 

7^  =  273°+   26°.l,  ai  =  0.1986, 

T2  =  273°  +  111°.3,  «2  =  0.9267. 

If  the  dissociation  takes  place  under  a  pressure  of  3  atmospheres, 
then  the  partial  pressures  of  the  component  gases  will  be 

1  -  a  2a 

-  '     and 


The  values  of  K^  and  Kp^  are,  then,  according  to  the  law  of 
mass  action  as  follows :  — 


,1  +  ail  4^ 

1  +  on 


276  THEORETICAL  CHEMISTRY 

and 

/  2tt2  \2 
U  +  az)          4a22 
P'          l-a2          l-«22' 

l+«2 
Substituting  in  equation  (6)  and  solving  for  Qp,  we  obtain 

4X(0.9267)2  4XC0.1986)2 

4.581  log  t  _  (Q  926?)2  -  log  x  _  (Q  1Q86)2  299.1  X  384.3 


p  299.1  -  384.3 

or 

Qp  =  -12,260  calories  per  mol  of  N204.     . 

In  a  reaction  which  is  accompanied  by  no  thermal  change, 
Q  =  0,  and  the  right-hand  side  of  equations  (1)  and  (2)  becomes 
equal  to  zero.  In  other  words,  in  such  a  reaction  a  change  in 
temperature  does  not  cause  a  displacement  of  the  equilibrium. 

The  reaction, 

C2H5OH  +  CH3COOH<=±  CH3COO.C2H5  +  H20, 

is  accompanied  by  such  a  small  thermal  change  that  it  may  be 
considered  as  zero,  and  according  to  the  above  reasoning  there 
should  be  only  a  very  slight  displacement  of  the  equilibrium  when 
the  temperature  is  varied.  Berthelot  found  that  at  10°  C., 
65.2  per  cent  of  the  alcohol  and  acid  are  changed  into  ester,  and 
at  220°  C.,  66.5  per  cent  of  the  mixture  is  transformed  into  ester. 
As  will  be  seen,  an  increase  of  210°  produces  hardly  any  displace- 
ment of  the  equilibrium. 

PROBLEMS. 

1.  When  2.94  mols  of  iodine  and  8.10  mols  of  hydrogen  are  heated  at 
constant  volume  at  444°  C.  until  equilibrium  is  established,  5.64  mols 
of  hydriodic  acid  are  formed.    If  we  start  with  5.30  mols  of  iodine  and 
7.94  mols  of  hydrogen,  how  much  hydriodic  acid  is  present  at  equilibrium 
at  the  same  temperature?  Ans.  9.49  mols. 

2.  At  2000°  C.,  and  under  atmospheric  pressure,  carbon  dioxide  is 
1.80  per  cent  dissociated  according  to  the  equation 


Calculate  the  equilibrium  constant  for  the  above  reaction  using  partial 
pressures.     ,  Ans.  3  X  10~6. 


HOMOGENEOUS  EQUILIBRIUM  277 

3.  What  is  the  equilibrium  constant  in  the  preceding  problem,  if  the 
concentrations  are  expressed  in  mols  per  liter?  Ans.   1.61  X  10~8. 

4.  When  6.63  mols  of  amylene  and  1  mol  of  acetic  acid  are  mixed, 
0.838  mol  of  ester  is  formed  in  the  total  volume  of  894  liters.     How  much 
ester  will  be  formed  when  we  start  with  4.48  mols  of  amylene  and  1  mol 
of  acetic  acid  in  the  volume  of  683  liters?  Ans.   0.8111  mol. 

5.  If  1  mol  of  acetic  acid  and  1  mol  of  ethyl  alcohol  are  mixed,  the 
reaction 

C2H5OH  +  CHaCOOH  <=±  CH3COOC2H5  +  H20, 

proceeds  until  equilibrium  is  reached,  when  £  mol  of  ethyl  alcohol,  £  mol 
of  acetic  acid,  f  mol  of  ethyl  acetate,  and  f  mol  of  water  are  present.  If 
we  start  (a)  with  1  mol  of  acid  and  2  mols  of  alcohol;  (b)  with  1  mol  of 
acid,  1  mol  of  alchol,  and  1  mol  of  water;  (c)  with  1  mol  of  ester  and  3 
mols  of  water,  how  much  ester  will  be  present  in  each  case  at  equilibrium? 
Ans.  (a)  0.845  mol,  (b)  0.543  mol,  (c)  0.465  mol. 

6.  In  the  reaction 


we  find,  since  I  0  =  i  02,  for 


the  values  3.02  at  386°  C.  and  2.35  at  419°  C.    Calculate  the  heat  evolved 
by  the  reaction  under  constant  pressure.  Ans.  6956  cal. 

7.  Above  150°  C.  N02  begins  to  dissociate  according  to  the  equation 


At  390°  C.  the  vapor  density  of  NO2  is  19.57  (H  =  1),  and  at  490°  C 
it  is  18.04.  Calculate  the  degree  of  dissociation  according  to  the  above 
equation  at  each  of  these  temperatures;  the  equilibrium  constants 
expressing  the  concentrations  in  mols  per  liter;  and  the  heat  of  dissoci- 
ation of  N02. 

Ans.  ttl  =  0.35,    az  =  0.55.     K^  =  2.884  X  10"2. 
Kz  =  7.173  X  10-2.  Q  =  -9160  cal. 


CHAPTER  XIII. 
HETEROGENEOUS  EQUILIBRIUM. 

Heterogeneous  Systems.  We  have  now  to  consider  equilibria 
in  systems  made  up  of  matter  in  different  states  of  aggregation. 
Such  systems  are  termed  heterogeneous  systems,  as  distinguished 
from  those  dealt  with  in  the  preceding  chapter  where  the  compo- 
sition is  uniform  throughout.  The  physically  distinct  portions  of 
matter  involved  in  a  heterogeneous  system  are  known  as  phases, 
each  phase  being  homogeneous  and  separated  from  the  other 
phases  by  definite  bounding  surfaces.  Thus,  ice,  liquid  water 
and  vapor  constitute  a  physically  heterogeneous  system.  Another 
heterogeneous  system  is  formed  by  calcium  carbonate  and  its 
dissociation  products,  calcium  oxide  and  carbon  dioxide.  The 
equilibrium  between  a  solid,  its  saturated  solution,  and  vapor 
affords  an  illustration  of  a  still  more  complex  heterogeneous 
system. 

Application  of  the  Law  of  Mass  Action  to  Heterogeneous 
Equilibria.  It  has  been  shown  in  the  preceding  chapter  that 
the  law  of  mass  action  may  be  applied  to  homogeneous  equilibria 
provided  the  molecular  condition  of  the  reacting  substances  is 
known. 

When  we  attempt  to  apply  the  law  of  mass  action  to  hetero- 
geneous equilibria,  especially  where  solids  are  involved,  the 
problem  presents  difficulties.  In  his  investigation  of  the  dis- 
sociation of  calcium  carbonate,  according  to  the  equation 


Debray  *  showed  that  just  as  every  liquid  has  a  definite  vapor 
pressure  corresponding  to  a  certain  temperature,  so  there  is  a 
definite  pressure  of  carbon  dioxide  over  calcium  carbonate  at  a 
definite  temperature.  Furthermore,  the  pressure  was  found  to 
be  independent  of  the  amount  of  calcium  carbonate  present. 

*  Compt.  rend.,  64,  603  (1867). 

278 


HETEROGENEOUS  EQUILIBRIUM  279 

Guldberg  and  Waage  *  showed  that  the  law  of  mass  action  can 
be  applied  to  such  heterogeneous  equilibria,  provided  that  the 
active  masses  of  the  solids  present  are  considered  as  constant. 

Nernst  pointed  out  that  this  statement  of  Guldberg  and  Waage 
can  be  easily  reconciled  with  experimental  facts.  In  a  hetero- 
geneous equilibrium  involving  solids,  it  is  only  necessary  to  con- 
sider the  gaseous  phase,  the  active  mass  of  a  solid  being  equivalent 
to  its  concentration  in  the  gaseous  phase.  That  is,  every  solid 
is  to  be  looked  upon  as  possessing,  at  a  definite  temperature,  a  defi- 
nite vapor  pressure  which  is  entirely  independent  of  the  amount  of 
solid  present.  Such  substances  as  arsenic,  antimony,  and  cadmium 
are  known  to  have  appreciable  vapor  pressures  at  relatively  low 
temperatures,  and  it  is  quite  reasonable  to  suppose  that  every 
solid  substance  exerts  a  definite  vapor  pressure  at  a  definite  temper- 
ature, even  though  we  have  no  method  sufficiently  refined  to  meas- 
ure such  minute  pressures. 

Since  the  active  mass  of  a  solid  remains  constant  so  long  as 
any  of  it  is  present,  the  application  of  the  law  of  mass  action 
to  certain  heterogeneous  equilibrium  is,  in  general,  simpler  than 
its  application  to  homogeneous  systems.  The  truth  of  this  state- 
ment will  be  evident  after  a  few  typical  heterogeneous  systems 
have  been  considered. 

(a)  Dissociation  of  Calcium  Carbonate.     In  the  reaction 
[CaC03]^±[CaO]  +  (C02), 

let  TTi  and  T2  represent  the  pressures  due  to  the  vapor  of  calcium 
carbonate  and  calcium  oxide  respectively,  and  let  p  denote  the 
pressure  of  the  carbon  dioxide.  Applying  the  law  of  mass  action, 
we  obtain 


But  since  v\  amd  KZ  are  constant  at  any  one  temperature,  the 
equation  becomes 

P  =  KP>, 

or,  the  equilibrium  constant  at  any  one  temperature  is  solely 
dependent  upon  the  pressure  of  the  carbon  dioxide  evolved.     The 

*  Loc.  cit. 


280 


THEORETICAL  CHEMISTRY 


accompanying  table  gives  the  values  of  the  pressure  of  carbon 
dioxide  corresponding  to  various  temperatures. 


Temperature, 
Degrees. 

Pressure  in 
Millimeters  of 
Mercury. 

547 

27 

610 

46 

625 

56 

740 

255 

745 

289 

810 

678 

812 

753 

865 

1333 

(b)  Dissociation  of  Ammonium  Hydrosulphide.  When  solid 
ammonium  hydrosulphide  is  heated,  it  is  almost  completely  dis- 
sociated into  ammonia  and  hydrogen  sulphide  as  shown  by  the 
following  equation:  — 

[NH4HS]^±(NH3) 


This  reaction  was  investigated  by  Isambert,*  who  found  that  the 
total  gas  pressure  at  25°.  1  C.  is  equal  to  501  mm.  of  mercury. 
Since  the  partial  pressures  of  the  ammonia  and  hydrogen  sulphide 
are  necessarily  the  same,  each  must  be  approximately  equal  to 
250.5  mm.,  the  relatively  small  pressure  due  to  the  undissociated 
vapor  of  the  ammonium  hydrosulphide  being  neglected.  Let  IT 
be  the  partial  pressure  of  the  vapor  of  ammonium  hydrosulphide, 
and  let  pi  and  p%  be  the  partial  pressures  of  the  ammonia  and 
hydrogen  sulphide.  Applying  the  law  of  mass  action,  we  have 

Pl'P2          K  m 

=  Ap-  (I) 


Since  TT  is  constant  at  any  one  temperature,  equation  (1)  becomes 

pi*pz  =  Kpf. 
According  to  Dalton's  law  of  partial  pressures,  we  have 

P  =   Pi  +  p2  +  7T, 

*  Compt.  rend.,  93,  595,  730  (1881). 


HETEROGENEOUS  EQUILIBRIUM 


281 


where  P  is  the  total  pressure.     Neglecting  the  relatively  small 
pressure  TT,  we  may  write 

P  =  Pi  +  P2- 

Hence,  since  pi  =  &, 


Substituting  these  values  in  equation  (1),  we  obtain 


The  value  of  the  equilibrium  constant  may  be  checked  by  observ- 
ing the  effect  on  the  system  of  the  addition  of  an  excess  of  either 
one  of  the  products  of  the  dissociation.  The  accompanying  table 
gives  the  results  of  a  few  of  Isambert's  experiments. 


Pressure  of 
Ammonia. 

Pressure  of 
Hydrogen 
Sulphide. 

PNH3*PH2S  =  KP- 

208 
138 
417 
453 

294 
458 
146 
143 

61,152 
63,204 
60,882 
64,779 

Mean  62,504 

As  will  be  seen,  the  mean  value  of  the  equilibrium  constant  agrees 
well  with  the  value  found  for  equivalent  amounts  of  the  products 
of  dissociation. 

(c)  Dissociation  of  Ammonium  Carbamate.  The  dissociation  of 
ammonium  carbamate  takes  place  according  to  the  equation 


XNH2 

This  dissociation  has  been  investigated  by  Horstmann.*    Applying 
the  law  of  mass  action,  we  have 

Pl2*ff2_   TT 

7T 

*  Lieb.  Ann.,  187,  48  (1877).! 


(1) 


282  THEORETICAL  CHEMISTRY 

where  pi  and  pz  are  the  partial  pressures  of  ammonia  and  carbon 
dioxide  respectively,  and  where  TT  is  the  partial  pressure  of  ammon- 
ium carbamate.  Since  TT  is  constant,  equation  (1)  becomes 

Kr. 


If  P  denotes  the  total  gaseous  pressure,  and  TT  is  neglected  as  in 
the  preceding  example,  we  have,  since  three  mols  of  gas  are 

formed 

P2  P 

Pi2=-g-,     and    P2=%- 

Substituting  these  values  in  equation  (1),  we  have 

p3   K, 

27      Kp' 

This  equation  has  also  been  tested  by  Isambert  *  by  adding  an 
excess  of  ammonia  or  carbon  dioxide  to  the  dissociating  system. 
He  found  that  the  value  of  the  equilibrium  constant  remains 
practically  constant.  The  addition  of  a  foreign  gas  was  shown 
to  be  without  effect  on  the  dissociation. 

(d)  Dissociation  of  the  Hydrates  of  Copper  Sulphate.  Many  inter- 
esting examples  of  heterogeneous  equilibrium  are  furnished  by  hy- 
drated  salts.  Thus,  if  crystallized  copper  sulphate,  CuS04.5  H2O, 
is  placed  in  a  desiccator,  it  gradually  loses  water  of  crystallization 
and  ultimately  only  the  anhydrous  salt  remains.  If  the  desiccator 
be  provided  with  a  manometer  and  is  so  arranged  that  the  tem- 
perature can  be  maintained  constant,  it  is  possible  to  observe  the 
changes  in  vapor  pressure  accompanying  the  process  of  dehydra- 
tion. At  the  temperature  of  50°  C.,  the  pressure  over  completely 
hydrated  copper  sulphate  is  found  to  remain  constant  at  47  mm. 
until  the  salt  has  been  deprived  of  two  molecules  of  water,  when 
it  drops  abruptly  to  30  mm.  and  remains  constant  until  two  more 
molecules  of  water  have  been  lost.  It  then  drops  again  to  4.4  mm. 
and  remains  constant  until  dehydration  is  complete. 

The  successive  stages  of  the  dehydration  are  shown  in  the  accom- 
panying diagram,  Fig.  71.  The  constant  pressures  observed  in 
the  dehydration  correspond  to  the  successive  equilibria  involved. 

*  Loc.  cit. 


HETEROGENEOUS  EQUILIBRIUM 


283 


At  50°  C.  the  pentahydrate  and  the  trihydrate  are  in  equilibrium, 
a  pressure  of  47  mm.  being  maintained  so  long  as  any  of  the  penta- 
hydrate is  present.  When  all  of  the  pentahydrate  is  used  up, 
then  the  trihydrate  begins  to  undergo  dehydration  into  the 
monohydrate.  This  is  a  new  equilibrium  and  the  pressure  of  the 


47  ram 


30mm 


4.5mm 


6H20 


3H2O 
Composition 

Fig.  71. 


1H20       OH20 


aqueous  vapor  necessarily  changes,  and  remains  constant  so  long 
as  any  trihydrate  remains.  The  last  stage  corresponds  to  the 
equilibrium  between  the  monohydrate  and  the  anhydrous  salt. 
The  following  equations  represent  the  three  successive  equilibria: — 

(1)  CuS04  •  5  H20  <=±  CuS04  •  3  H20  +  2  H20, 

(2)  CuS04.3H20^±CuS04-H20  +  2H20, 

(3)  CuS04  •  H20  <=±  CuS04  +  H20. 

Applying  the  law  of  mass  action  to  the  first  of  the  above  equi- 
libria, we  have 

Trip2      v 

-  =  Ap, 

7T2 

in  which  TTI  and  7r2  denote  the  partial  pressures  due  to  the  hydrates 
CuSO4.5  H20  and  CuS04.3  H20  respectively,  and  p  denotes  the 


284 


THEORETICAL  CHEMISTRY 


pressure  of  aqueous  vapor.  Since  TTI  and  TTZ  are  constant,  the 
above  expression  simplifies  to  the  following 

p2  =  Kp'. 

In  a  similar  manner  it  may  be  shown  that  the  pressure  of  aqueous 
vapor  in  the  other  equilibria  must  be  constant.  It  must  be 
clearly  understood  that  the  observed  pressure  is  only  definite 
and  fixed  when  two  hydrates  are  present.  If  the  dehydration 


Ice 


Temperature 
Fig.  72. 

were  conducted  at  another  temperature  than  50°  C.  the  equilibrium 
pressure  would  be  different.  The  vapor  pressure  curves  of  the 
different  hydrates  are  shown  in  the  temperature-pressure  diagram 
of  Fig.  72. 

Heat  of  Dissociation  of  Solids.  When  the  products  of  the 
dissociation  of  a  solid  are  gaseous,  it  has  been  pointed  out  by  De 
Forcrand  *  that  the  ratio  of  the  heat  of  dissociation  of  1  mol  of 

*  Ann.  Chim.  Phys.  [7],  28,  545. 


HETEROGENEOUS  EQUILIBRIUM 


285 


solid  to  the  absolute  temperature  at  which  the  dissociation  pres- 
sure is  equal  to  1  atmosphere,  is  constant.  Or,  denoting  the  heat 
of  dissociation  by  Q  and  the  absolute  temperature  by  T,  De  For- 
crand's  relation  may  be  expressed  thus, 

^  =  constant  =  33. 

Nernst  has  shown  that  the  value  of  the  constant  in  this  relation 
is  not  independent  of  the  temperature.  Thus,  the  value  of  the 
ratio  at  100°  C.  is  29.7,  while  at  1000°  C.  it  is  37.7.  Up  to  the 
present  time  no  expression  has  been  derived  in  which  the  variation 
of  the  ratio  with  the  temperature  is  included. 

Distribution  of  a  Solute  between  Two  Immiscible  Solvents. 
When  an  aqueous  solution  of  succinic  acid  is  shaken  with  ether, 
the  acid  distributes  itself  between  the  ether  and  the  water  in  such 
a  way  that  the  ratio  between  the  two  concentrations  is  always 
constant.  It  will  be  seen  that  the  distribution  of  the  succinic 
acid  between  the  two  solvents  is  analogous  to  that  of  a  substance 
between  the  liquid  and  gaseous  phases  (see  page  148),  and  there- 
fore the  laws  governing  the  latter  equilibrium  should  apply  equally 
to  the  former.  Nernst  *  has  shown  that  (a)  //  the  molecular 
weight  of  the  solute  is  the  same  in  both  solvents,  the  ratio  in  which  it 
distributes  itself  between  them  is  constant  at  constant  temperature, 
or  in  other  words,  Henry's  law  is  applicable;  and  (b)  //  there  are 
several  solutes  in  solution  the  distribution  of  each  solute  is  the  same  as 
if  it  were  present  alone.  This  is  clearly  Dalton's  law  of  partial 
pressures.  The  ratio  in  which  the  solute  distributes  itself  between 
the  two  solvents  is  termed  the  coefficient  of  distribution  or  partition. 
The  following  table  gives  the  results  of  three  experiments  on  the 
distribution  of  succinic  acid  between  ether  and  water. 


Concentration 
in  Water. 

Concentration 
in  Ether. 

Distribution 
Coefficient. 

43.4 
43.8 
47.4 

7.1 
7.4 

7.9 

6.1 
5.9 
6.0 

Zeit.  phys.  Chem.,  8,  110  (1891). 


286 


THEORETICAL  CHEMISTRY 


As  will  be  seen  the  distribution  coefficient  is  constant,  showing 
that  Henry's  law  applies.  When  the  molecular  weight  of  one 
solute  is  not  the  same  in  both  solvents  the  distribution  coefficient 
is  not  constant,  and  conversely,  if  the  distribution  coefficient  is 
not  constant,  we  infer  that  the  molecular  weights  of  the  solute 
in  the  two  solvents  are  not  identical. 

Let  us  assume  that  a  solute  whose  normal  molecular  weight  is 
A,  when  shaken  with  two  immiscible  solvents  undergoes  polymeri- 
zation in  one  of  them,  its  molecular  weight  being  An.  We  then 
have  the  equilibrium 

An  <=±  nA  : 

applying  the  law  of  mass  action,  we  have 


If  the  molecular  weight  in  one  solvent  is  twice  the  molecular 
weight  in  the  other,  then  n  =  2,  and 


=  constant. 


Thus  Nernst  found  the  following  concentrations  of  benzoic  acid 
when  it  was  shaken  with  benzene  and  water. 


ci  (Water). 

c2  (Benzene). 

£1 

C2' 

%• 

0.0150 
0.0195 
0.0289 

0.242 
0.412 
0.970 

0.062 
0.048 
0.030 

0.0305 
0.0304 
0.0293 

As  will  be  seen,  the  values  of  the  ratio  Ci/c2  steadily  decrease, 
while  on  the  other  hand,  the  values  of  the  ratio  Ci/Vcz  remain 
constant.  This  shows,  therefore,  that  benzoic  acid  has  twice  the 
normal  molecular  weight  in  benzene. 

The  Solution  of  a  Solid  in  a  Non-dissociating  Solvent.  When 
a  solid  is  brought  in  contact  with  a  non-dissociating  solvent,  it 
continues  to  dissolve  until  the  solution  becomes  saturated.  A 
condition  of  equilibrium  then  obtains,  the  rates  of  solution  and 


HETEROGENEOUS  EQUILIBRIUM  287 

precipitation  being  the  same.  This  is  plainly  a  case  of  hetero- 
geneous equilibrium.  If  c  is  the  concentration  of  the  dissolved 
substance,  and  TT  is  the  concentration  of  the  undissolved  solid,  then 
according  to  the  law  of  mass  action 


or  since  TT  is  constant, 

c  =  Kc'. 

Variation  of  the  Constant  of  Heterogeneous  Equilibrium  with 
Temperature.     The  reaction  isochore  equation  of  Van't  Hoff 

d  Qog  JQ       J2_ 
dT          RT2' 

which  has  been  shown  to  connect  the  displacement  of  a  homo- 
geneous equilibrium  with  change  in  temperature,  applies  equally 
well  to  heterogeneous  equilibria.  The  following  examples  will 
serve  to  illustrate  its  application  in  such  cases. 

(a)  Dissociation  of  Ammonium  Hydrosulphide.    In  the  reaction 
representing  the  dissociation  of  ammonium  hydrosulphide, 


let  pi  and  p2  be  the  partial  pressures  of  ammonia  and  hydrogen 
sulphide,  and  let  TT  be  the  partial  pressure  of  ammonium  hydro- 
sulphide.  Then  as  has  been  shown  (see  page  281), 

K*'  -  -4  > 

where  P  is  the  total  gaseous  pressure.     From  the  following  data:  — 

Ti  =  273°  +  9°.5,  Pi  =  175  mm.  of  mercury, 

and 

T2  =  273°  +  25°.l,  P2  =  501  mm.  of  mercury, 

we  have,  on  applying  the  reaction  isochore  equation,  and  solving 
for  Qp, 


4.581  ["log  (n^Y  ~  log  (^r)2]  282.5  X  298.1 
Qp=  282.5  -  298.1 

or  OP  =  -  22,740  calories. 


288  THEORETICAL  CHEMISTRY 

This  result  agrees  well  with  the  value,  —  22,800  calories,  found 
by  direct  experiment. 

(b)  Solution  of  Succinic  Acid.    The  concentration  of  succinic 
acid  (in  a  saturated  solution)  and  the  temperature,  are  the  factors 
which  determine  the  equilibrium  in  this  case.     In  the  equation 
d  (logegc)  _    Qe 


dT  RT2 

Kc  =  c,  where  c  is  the  concentration  of  succinic  acid  in  a  saturated 
solution.  The  following  experimental  data,  due  to  Van't  Hoff, 
enables  us  to  calculate  the  heat  of  solution  of  the  acid. 

Ti  =  273°  ci  =  2.88  mols  per  liter, 

and 

T2  =  273°  +  8°.5,         02  =  4.22  mols  per  liter. 

Substituting  in  the  reaction  isochore  equation  and  solving  for  Qc, 
we  have 

n       4.581  (log  4.22  -  log  2.88)  273  X  281  .5 

^c  =  273  -  281.5 

or 

Qc  =  -6900  calories. 

The  value  of  the  heat  of  solution  for  1  mol  of  succinic  acid  as 
found  by  direct  experiment  is  —6700  calories. 

The  Phase  Rule.  While  it  is  possible  to  apply  the  law  of 
mass  action  to  certain  heterogeneous  equilibria  there  are  numerous 
cases  where  its  application  is  either  difficult  or  impossible.  To 
deal  with  such  heterogeneous  systems  we  make  use  of  a  general- 
ization discovered  by  J.  Willard  Gibbs,*  late  professor  of  mathe- 
matical physics  in  Yale  University.  This  generalization  was  first 
stated  by  Gibbs  in  1874,  and  is  commonly  known  as  the  phase  rule. 
Before  entering  upon  a  discussion  of  the  phase  rule,  it  will  be 
necessary  to  define  a  few  of  the  terms  employed. 

The  composition  of  a  system  is  determined  by  the  number  of 
independent  variables  or  components  involved.  Thus  in  the 
system  —  ice,  water,  and  vapor  —  there  is  but  a  single  com- 
ponent. In  the  system 


*  Trans.  Connecticut  Academy,  Vols.  II  and  III,  1875-8. 


HETEROGENEOUS  EQUILIBRIUM  289 

while  there  are  three  constituents  of  the  equilibrium,  only  two  of 
these  need  be  considered  as  components,  for  the  amount  of  any 
one  constituent  is  not  independent  of  the  amounts  of  the  other 
two,  as  the  following  equations  show :  — 

CaO  +  C02  =  CaC03, 

CaCO3  -  CaO  =  C02, 

CaC03  -  C02  =  CaO. 

In  general,  the  components  are  chosen  from  the  smallest  number 
of  independently-variable  constituents  required  to  express  the 
composition  of  each  phase  entering  into  the  equilibrium,  even 
negative  quantities  of  the  components  being  permissible. 

The  number  of  variable  factors,  —  temperature,  pressure,  and 
concentration, —  of  the  components  which  must  be  arbitrarily  fixed 
in  order  to  define  the  condition  of  the  system,  is  known  as  the 
degree  of  freedom  of  the  system.  For  example,  a  gas  has  two 
degrees  of  freedom  since  two  of  the  variables,  temperature,  pres- 
sure or  volume,  must  be  fixed  in  order  to  define  it;  a  liquid  and  its 
vapor  has  only  one  degree  of  freedom,  since  for  equilibrium  at  a 
certain  temperature,  there  can  be  but  a  single  pressure;  and  in  a 
system  consisting  of  a  substance  in  the  three  states  of  aggregation, 
equilibrium  can  only  exist  at  a  single  temperature  and  pressure. 

Derivation  of  the  Phase  Rule.  The  following  derivation  of 
the  phase  rule  is  due  to  Nernst.  Let  us  assume  a  complete  hetero- 
geneous equilibrium  made  up  of  y  phases  of  n  components,  and  let 
us  fix  our  attention  upon  one  single  phase.  This  phase  will  con- 
tain a  certain  amount  of  each  one  of  the  n  components,  the  con- 
centrations of  which  may  be  designated  by  Ci,  C2,  Ca,  .  .  .  cn. 
Since  we  have  assumed  complete  equilibrium  to  exist,  the  slightest 
change  in  concentration,  temperature  or  pressure  will  alter  the 
composition  of  this  phase. 

This  may  be  expressed  by  the  equation 

/  (ci,  02,  c3,  .  .  .  cn,  p,  T)  =  0, 

where  /  is  any  function  of  the  variables.  Since  any  change  in 
one  phase  implies  a  corresponding  change  in  the  remaining  y  —  1 
phases,  it  follows  that  the  composition  of  all  the  phases  is  a  certain 
determined  function  of  the  same  variables. 


290  THEORETICAL  CHEMISTRY 

The  above  equation  is,  then,  of  the  form  ascribed  to  each  sepa- 
rate phase,  and  since  there  are  y  phases  we  have  y  separate  equa- 
tions. There  are,  however,  n  +  2  variables  in  each  equation,  so 
that  if  y  =  n  +  2,  that  is  if  we  have  two  more  phases  than  com- 
ponents, each  unknown  quantity  has  a  definite  known  value. 
In  this  case  there  is  only  one  value  for  ci,  C2,  c3,  c4,  .  .  .  cn,  p  and 
T  at  which  the  system  can  be  in  equilibrium.  Hence  when  n 
components  are  present  in  n  +  2  phases,  we  have  equilibrium  only 
for  a  certain  temperature,  a  certain  pressure,  and  a  certain  ratio 
of  concentrations  of  the  single  phases.  That  is,  n  +  2  phases  of  n 
substances  can  only  exist  at  a  certain  point  in  a  co-ordinate 
system.  This  point  is  termed  the  transition  point.  If  one  value 
be  altered  when  one  phase  vanishes,  and  there  remain  n  +  1  phases 
of  n  components,  the  problem  becomes  indeterminate.  Thus 
it  is  proved  that  n  components  are  necessary  in  order  that  a  system 
containing  n  +  1  phases  may  exist  in  complete  equilibrium. 

The  phase  rule  may  be  stated  as  follows:  —  A  system  made  up 
of  n  components  in  n  +  2  phases  can  only  exist  when  pressure, 
temperature  and  concentration  have  definite  fixed  values;  a  system 
of  n  components  in  n  +  1  phases  can  exist  only  so  long  as  one  of  the 
factors  varies;  and  a  system  of  n  components  in  n  phases  can  exist 
only  so  long  as  two  of  the  factors  vary.  If  P  denotes  the  number  of 
phases,  C  the  number  of  components,  and  F  the  number  of  degrees 
of  freedom,  then  the  phase  rule  may  be  conveniently  summarized 
by  the  expression, 

C  -  P  +  2  =  F. 

Equilibrium  in  the  System,  Water,  Ice,  and  Vapor.  In  this 
system  we  may  have  one,  two,  or  three  phases  present,  according 
to  the  conditions.  Under  ordinary  circumstances  of  temper- 
ature and  pressure,  water  and  water  vapor  are  in  equilibrium. 
The  vapor  pressure  curve  of  water  is  represented  by  the  line  OA 
in  the  pressure-temperature  diagram  (Fig.  73).  It  is  only  at 
points  on  this  curve  that  water  and  its  vapor  are  in  equilibrium. 
Thus,  if  the  pressure  be  reduced  below  that  corresponding  to  any 
point  on  OA,  all  of  the  water  will  be  vaporized;  if  on  the  other 
hand,  the  pressure  be  raised  above  the  curve,  all  of  the  vapor  will 


HETEROGENEOUS  EQUILIBRIUM 


291 


ultimately  condense  to  the  liquid  state.  When  the  temperature 
is  reduced  below  0°  C.,  only  ice  and  vapor  are  present,  the  curve 
OC  representing  the  equilibrium  between  these  two  phases.  It  is 
to  be  observed  that  the  curve  OC  is  not  continuous  with  OA .  At 


Solid 


Vapor 


0.0075 

Temperature 
Fig.  73. 

the  point  0,  where  the  two  curves  intersect,  ice,  water,  and  water 
vapor  are  in  equilibrium.  At  this  point  ice  and  water  must  have 
the  same  vapor  pressure,  otherwise  distillation  of  vapor  from  the 
phase  having  the  higher  vapor  pressure  to  that  with  the  lower 
vapor  pressure  would  occur,  and  eventually  the  phase  having  the 
higher  vapor  pressure  would  disappear.  This  result  would  be 
in  contradiction  to  the  experimentally-determined  fact  that 
both  solid  and  liquid  phases  are  in  equilibrium  at  the  point  0. 
The  temperature  at  which  ice  and  water  are  in  equilibrium  with 
their  vapor  under  atmospheric  pressure  is  0°  C.  Since  increase 
of  pressure  lowers  the  freezing-point  of  water,  the  point  0,  repre- 
senting the  equilibrium  between  ice  and  water  under  the  pressure 
of  their  own  vapor,  viz.,  4.57  mm.,  must  be  a  little  above  0°  C. 
The  exact  temperature  corresponding  to  the  point  0  has  been 
found  to  be  O.°0075  C. 


292  THEORETICAL  CHEMISTRY 

The  change  in  the  melting-point  of  ice  due  to  increasing  pressure 
is  represented  by  the  line  OB.  This  line  is  inclined  toward  the 
vertical  axis  because  the  melting  point  of  ice  is  lowered  by  in- 
creased pressure.  The  point  0  is  called  a  triple  point  because 
there,  and  there  only,  three  phases  are  in  equilibrium.  As  is  well 
known,  water  does  not  always  freeze  exactly  at  0°  C.  If  the 
containing  vessel  is  perfectly  clean,  and  care  is  taken  to  exclude 
dust,  it  is  possible  to  supercool  water  several  degrees  below  its 
freezing-point  and  measure  its  vapor  pressure. 

The  dotted  curve  OAf,  which  is  a  continuation  of  OA,  represents 
the  vapor  pressure  of  supercooled  water.  It  will  be  noticed  that 
(1)  there  is  no  break  in  the  vapor-pressure  curve  so  long  as  the 
solid  phase  does  not  separate,  and  (2)  the  vapor  pressure  of  super- 
cooled water,  which  is  an  unstable  phase,  is  greater  than  that  of 
ice,  the  stable  phase,  at  that  temperature. 

We  now  proceed  to  apply  the  phase  rule  to  this  system.  In  the 
formula,  C  -  P  +  2  =  F,  C  =  1.  It  is  evident  that  if  P  =  3, 
then  F  —  0;  or  the  system  has  no  degree  of  freedom.  We  have 
seen  that  the  triple  point  0,  represents  such  a  condition.  At  this 
point  ice,  water,  and  water  vapor  are  co-existent,  and  if  either 
one  of  the  variables,  temperature  or  pressure,  is  altered,  one  of 
the  phases  disappears;  in  other  words,  the  system  has  no  degree 
of  freedom.  Such  a  system  is  said  to  be  non-variant.  If  in  the 
above  formula,  P  =  2,  then  F  =  1,  and  the  system  has  one  degree 
of  freedom,  or  is  univariant.  Any  point  on  any  one  of  the  curves 
OA,  OB,  or  OC  represents  a  univariant  system.  Take,  for  exam- 
ple, a  point  on  the  curve  OA.  In  this  case  the  temperature  may 
be  altered  without  altering  the  number  of  phases  in  equilibrium. 
If  the  temperature  is  raised,  a  corresponding  increase  in  vapor 
pressure  follows  and  the  system  will  adjust  itself  to  some  other 
point  on  the  curve  OA.  In  like  manner,  the  pressure  may  be 
altered  without  causing  the  disappearance  of  one  of  the  phases. 
If,  however,  the  temperature  is  maintained  constant,  then  a  change 
in  the  pressure  will  cause  either  condensation  of  water  vapor  or 
vaporization  of  liquid  water.  Under  these  conditions  the  system 
has  only  one  degree  of  freedom.  Again,  if  P  =  1,  then  F  =  2, 
and  the  system  is  Invariant,  or  has  two  degrees  of  freedom.  The 


HETEROGENEOUS  EQUILIBRIUM  293 

areas  included  between  the  curves  in  the  diagram  are  examples 
of  bivariant  systems.  Consider  the  vapor  phase;  the  temperature 
may  be  fixed  at  any  desired  value  within  the  vapor  area  AOC, 
and  the  pressure  may  be  altered  along  a  line  parallel  to  the  vertical 
axis  without  causing  a  change  in  the  number  of  phases,  provided 
the  curves  OA  and  OC  are  not  intersected. 

The  System,  Sulphur  (Rhombic,  Monoclinic),  Liquid  and 
Vapor.  This  system  is  more  complicated  than  the  preceding 
one-component  system,  since  there  are  two  solid  phases  in  addition 
to  the  liquid  and  vapor  phases.  At  ordinary  temperatures,  rhom- 
bic sulphur  is  the  stable  modification.  When  this  is  heated 
rapidly  it  melts  at  115°  C.,  but  if  it  is  maintained  in  the  neighbor- 
hood of  100°  C.  it  gradually  changes  into  monoclinic  sulphur 
which  melts  at  120°  C.  Monoclinic  sulphur  can  be  kept  indefin- 
itely at  100°  C.  without  undergoing  change  into  the  rhombic 
modification,  or  in  other  words  it  is  the  stable  phase  at  this  temper- 
ature. 

It  is  evident,  therefore,  that  there  must  be  a  temperature  above 
which  monoclinic  sulphur  is  the  stable  form  and  below  which 
rhombic  sulphur  is  the  stable  modification.  This  temperature 
at  which  both  rhombic  and  monoclinic  modifications  are  in  equi- 
librium with  each  other  and  with  their'  vapor,  is  termed  the 
transition  point.  Its  value  has  been  determined  to  be  95°.6  C. 
The  change  from  one  form  into  the  other  is  relatively  slow,  so 
that  it  is  possible  to  measure  the  vapor  pressure  of  rhombic  sul- 
phur up  to  its  melting-point,  and  that  of  monoclinic  sulphur 
below  its  transition  point.  The  vapor  pressure  of  solid  sulphur, 
although  very  small,  has  been  measured  as  low  as  50°  C. 

The  complete  pressure-temperature  diagram  for  sulphur  is 
shown  in  Fig.  74.  At  the  point  0,  rhombic  and  monoclinic  sul- 
phur are  in  equilibrium  with  sulphur  vapor,  this  being  a  triple 
point  analogous  to  the  point  0  in  Fig.  73.  The  vapor  pressure 
curves  of  rhombic  and  monoclinic  sulphur  are  represented  by  OB 
and  OA  respectively.  The  dotted  curve  OA'  which  is  a  continu- 
ation of  OA  is  the  vapor-pressure  curve  of  monoclinic  sulphur  in 
a  metastable  region.  In  like  manner  OB'  represents  the  vapor- 
pressure  curve  of  rhombic  sulphur  in  the  metastable  condition,  B' 


294 


THEORETICAL  CHEMISTRY 


being  a  metastable  melting  point.  As  in  the  pressure-temper- 
ature diagram  for  water,  the  metastable  phases  have  the  higher 
vapor  pressures.  The  effect  of  increasing  pressure  on  the  transi- 
tion point  0,  is  represented  by  the  line  OC.  This  is  termed  a 


^Rhombic  Sulphur 


Vapor 


Temperature 
Fig.  74. 

transition  curve,  and,  since  increase  in  pressure  raises  the  transi- 
tion point,  the  line  slopes  away  from  the  vertical  axis.  The 
effect  of  increased  pressure  on  the  melting-point  of  monoclinic 
sulphur  is  shown  by  the  curve  AC. 

This  also  slopes  away  from  the  vertical  axis,  but  the  change  in 
the  melting-point  of  monoclinic  sulphur  produced  by  a  given 
change  in  pressure  being  less  than  the  corresponding  change  in  the 


HETEROGENEOUS  EQUILIBRIUM  295 

transition  point,  the  two  curves,  OC  and  AC,  intersect  at  the 
point  C.  The  point  C  corresponds  to  a  temperature  of  131°  C. 
and  a  pressure  of  400  atmospheres.  The  vapor-pressure  curve 
of  stable  liquid  sulphur  is  represented  by  the  curve  AD.  The 
vapor-pressure  curve  of  the  metastable  liquid  phase  is  represented 
by  the  curve  AB'  which  is  continuous  with  AD.  The  diagram  is 
completed  by  the  curve  B'C  which  represents  the  effect  of  pressure 
on  the  metastable  melting-point  of  rhombic  sulphur.  Mono- 
clinic  sulphur  does  not  exist  above  the  point  C;  hence  when 
liquid  sulphur  is  allowed  to  solidify  at  pressures  exceeding  400  at- 
mospheres, the  rhombic  modification  is  formed,  whereas  under 
ordinary  pressures  the  monoclinic  modification  appears  first. 

The  phase  rule  enables  us  to  state  the  exact  conditions  required 
for  equilibrium  in  this  system  and  to  check  the  results  of  exper- 
iment. Thus,  according  to  the  formula,  C  —  P  +  2  =  F,  since 
C  =  1,  the  system  will  be  non- variant  when  P  =  3.  Since  there 
are  four  phases  involved,  theoretically  any  three  of  these  may 
be  co-existent  and  four  triple  points  are  possible.  The  theoreti- 
cally-possible triple  points  are  as  follows:  — 

(1)  Rhombic  sulphur,  monoclinic  sulphur,  and  vapor  (0); 

(2)  Rhombic  sulphur,  monoclinic  sulphur,  and  liquid  (C); 

(3)  Rhombic  sulphur,  liquid,  and  vapor  (B') ; 

(4)  Monoclinic  sulphur,  liquid  and  vapor  (A). 

In  this  particular  system  all  of  the  four  possible  triple  points  can 
be  realized  experimentally.  That  this  is  the  case  is  due  to  the 
comparative  slowness  of  the  change  from  rhombic  to  monoclinic 
sulphur  above  the  triple  point.  If  this  change  were  rapid  it  is 
evident  that  all  of  the  theoretically-possible  non-variant  systems 
could  not  be  realized  experimentally. 

As  in  the  case  of  water,  the  curves  in  the  diagram  represent 
univariant  systems  and  the  areas  bivariant  systems.  The  student 
is  advised  to  tabulate  the  univariant  and  bivariant  systems  repre- 
sented in  the  pressure-temperature  diagram  for  sulphur. 

Two-component  Systems.  Turning  now  to  two-component 
systems  we  are  confronted  with  a  more  difficult  problem,  and  one 
which  includes  many  special  cases.  Thus,  we  may  have  cases  of 


296 


THEORETICAL  CHEMISTRY 


anhydrous  salts  and  water,  hydrated  salts  and  water,  volatile 
solutes,  two  liquid  phases,  consolute  liquids,  and  solid  solutions. 
To  enter  upon  a  discussion  of  these  would  not  be  profitable,  since 
they  only  serve  to  give  greater  emphasis  to  the  general  truth  of 
the  phase  rule.  We  shall  select  a  few  typical  two-component 
systems  for  consideration  here. 

(a)  Anhydrous  Salt  and  Water.     In  the  equilibrium  diagram 
of  water  (here  represented  by  dotted  lines,  Fig.  75),  we  desig- 


Temperature 
Fig.  75. 

nate  the  triple  point  by  0.  At  this  point  ice  and  water  have  the 
same  vapor  pressure.  Similarly,  a  solution  at  its  freezing-point 
has  the  same  vapor  pressure  as  the  ice  which  separates.  The 
intersection  of  the  vapor-pressure  curve  for  ice,  OB,  and  the  vapor- 
pressure  curve  of  the  solution  of  the  anhydrous  salt,  0"A",  deter- 
mines a  new  triple  point  0".  Since  the  presence  of  the  dissolved 
salt  tends  to  diminish  the  vapor  pressure  of  water,  the  curve  0"A" 
is  situated  below  the  curve  OA,  and  for  the  same  reason  the  triple 
point  0"  is  found  to  the  left  of  0.  If  now  we  keep  an  excess  of 
dissolved  substance  continually  present,  all  of  the  liquid  phases 


HETEROGENEOUS  EQUILIBRIUM  297 

which  are  formed  will  of  necessity  be  saturated  solutions.  When 
these  solutions  finally  freeze  they  will  furnish,  not  pure  ice,  but  a 
mixture  of  ice  and  solid  salt,  known  as  a  cryohydrate.  By  a 
partial  freezing  we  can  therefore  obtain  the  system:  Solid  salt, 
ice,  saturated  solution  and  vapor,  or  in  other  words,  a  system  of 
n  +  2  phases  of  which  the  existence  is  only  possible  at  the  freez- 
ing temperature  Tf  of  the  saturated  solution,  and  under  the  pres- 
sure pr  corresponding  to  the  vapor  pressure  of  ice  and  the  saturated 
solution.  These  conditions  are  represented  in  the  diagram  by 
the  quadruple  point  0'.  If  now  we  pass  from  the  point  0',  increas- 
ing the  temperature  and  pressure  as  prescribed  by  the  curve 
O'A',  the  ice  disappears,  while  the  salt,  the  saturated  solution, 
and  the  vapor  furnish  a  series  of  3-phase  systems.  Again  start- 
ing from  the  point  0'  and  lowering  the  temperature  and  the  pres- 
sure as  indicated  by  the  curve  O'B,  the  liquid  phase  disappears, 
while  the  solid  salt,  ice,  and  vapor  constitute  another  series  of 
3-phase  systems.  This,  of  course,  is  on  the  supposition  that  the 
vapor  pressure  of  the  solid  salt  is  negligible.  Finally,  a  consider- 
able increase  in  pressure  causes  a  slight  lowering  of  the  temper- 
ature corresponding  to  the  quadruple  point,  the  conditions  being 
represented  by  the  curve  O'C'. 

All  possible  non-saturated  solutions  of  the  salt  will  be  repre- 
sented by  points  within  the  area,  AOO'A'.  Thus,  let  0"A"  repre- 
sent the  vapor-pressure  curve  of  a  dilute  solution  of  the  salt  in 
water.  The  freezing-point  of  this  solution  is  represented  by  the 
point  0",  while  0"C"  represents  the  variation  of  the  freezing-point 
of  the  solution  with  pressure. 

The  following  table  summarizes  the  possibilities  indicated  by 
the  phase  rule :  — 

4  phases;  salt,  ice,  saturated  solution,  vapor  (point,  0'); 

3  phases;  salt,  saturated  solution,  vapor  (curve,  O'A'); 

3  phases;  salt,  ice,  vapor  (curve,  O'B)', 

3  phases;  salt,  ice,  saturated  solution  (curve,  O'C'); 

2  phases;  salt,  saturated  solution  (area,  00' 'A') ; 

2  phases;  salt,  water  vapor  (area,  BO'A'}\ 

2  phases;  salt,  ice  (area,  BO'C'Y, 


298 


THEORETICAL  CHEMISTRY 


3  phases;  ice,  non-saturated  solution,  vapor  (curve,  00'); 

2  phases;  non-saturated  solution,  vapor  ) ,  if\r\t  A  /\ 

{(area,  AOO'A')', 

1  phase;  non-saturated  solution  ) 

2  phases;  non-saturated  solution,  ice        )  ,          CGO'C'} 
1  phase;  non-saturated  solution  ) 

As  will  be  seen,  there  is  only  one  non-variant  point  in  the  entire 
diagram,  viz.,  the  point  0'.  In  this  system  there  are  three  degrees 
of  freedom,  since  in  addition  to  temperature  and  pressure  the  con- 
centration of  the  solution  may  also  be  varied. 

The  pressure-temperature  diagram  (Fig.  75)  having  been  dis- 
cussed, we  now  turn  to  the  concentration-temperature  diagram 
for  the  same  system,  Fig.  76.  In  this  diagram  the  abscissae 


Temperature 
Fig.  76. 

represent  temperatures  and  the  ordinates,  concentrations.  For 
convenience,  corresponding  points  in  Figs.  75  and  76  will  be  desig- 
nated by  the  same  letters.  The  equilibrium  between  ice,  water 
and  water  vapor  is  represented  by  the  point  0.  If  now  a  small 


HETEROGENEOUS  EQUILIBRIUM  299 

amount  of  anhydrous  salt  be  added  to  the  water,  the  freezing-point 
will  be  lowered  to  0" .  As  the  proportion  of  salt  is  increased  the 
temperature  of  equilibrium  is  lowered  along  the  curve  00"0f.  A 
point  is  ultimately  reached  at  which  the  solution  becomes  saturated, 
and  on  further  addition  of  salt  it  is  not  dissolved,  but  remains  in 
contact  with  the  ice  and  saturated  solution.  This  is  the  cryo- 
hydric  point,  and  represents  the  lowest  temperature  which  can  be 
obtained  in  this  particular  system.  The  diagram  is  completed 
by  the  solubility  curve  of  the  salt,  O'A'.  Each  point  on  this 
curve  represents  the  concentration  of  the  saturated  solution  at 
all  temperatures,  from  the  critical  temperature  of  the  solution  to 
the  cryohydric  temperature.  The  meaning  of  the  concentration- 
temperature  diagram  may  be  made  clearer  by  a  consideration  of 
the  behavior  of  a  solution  when  gradually  cooled.  Let  a  repre- 
sent a  dilute  solution  of  the  anhydrous  salt.  On  lowering  the 
temperature  along  ab,  no  change  will  occur  until  the  curve  00' 
is  reached;  then  ice  will  begin  to  separate  and  as  the  cooling  is 
continued,  the  composition  of  the  solution  will  change  along  00' 
until  it  reaches  the  cryohydric  point  0'.  Here  both  salt  and  ice 
will  separate,  and  the  solution  will  solidify  completely  at  the 
temperature  corresponding  to  the  point  0'.  In  like  manner,  if 
we  start  with  a  concentrated  solution  represented  by  the  point  c 
and  cool  along  cd  no  change  will  take  place  until  the  curve  O'A' 
is  reached;  then  solid  salt  will  separate  and  the  composition  of 
the  solution  will  alter  along  O'A'  until  the  temperature  is  reduced 
to  that  corresponding  to  the  cryohydric  point,  when  the  whole 
solution  will  solidify  as  in  the  previous  case.  This  phenomenon 
was  first  systematically  investigated  by  Guthrie  *  who  concluded 
that  such  mixtures  of  constant  composition  and  definite  melting- 
point  are  chemical  compounds,  and,  therefore,  he  proposed  to  call 
them  cryohydrates.  It  has  since  been  shown  that  cryohydrates 
are  not  definite  chemical  compounds.  Among  the  various  reasons 
which  have  been  advanced  to  prove  the  incorrectness  of  Guthrie's 
views,  the  following  are  the  most  cogent: — (1)  the  physical 
properties  of  a  cryohydrate  are  the  mean  of  the  corresponding 
properties  of  the  constituents,  this  being  rarely  true  of  chemical 
*  Phil.  Mag,  [4],  49,  1  (1875);  [5],  i,  49  and  2,  211  (1876). 


300 


THEORETICAL  CHEMISTRY 


compounds;  (2)  the  lack  of  homogeneity  of  a  cryohydrate  can 
be  detected  under  the  microscope;  and  (3)  the  constituents  are 
seldom  present  in  simple  molecular  proportions. 

Applying  the  phase  rule  to  the  above  two-component  system, 
it  is  evident  that  there  is  but  one  non-variant  system:  this  is 
represented  by  the  point  0'.  When  three  phases  are  co-existent 
the  system  is  univariant,  when  only  two  phases  are  present  the 
system  is  bivariant,  and  finally,  when  only  one  phase  is  present 
the  system  acquires  three  degrees  of  freedom  or  is  trivariant. 
It  is  evident  that  a  system  having  three  degrees  of  freedom  cannot 
be  completely  represented  by  a  diagram  in  a  single  plane.  It  is 
possible,  however,  to  construct  a  three-dimensional  model  which 
will  represent  the  equilibrium  very  satisfactorily.  Such  a  model  is 


Fig.  77. 

I.  Unsaturated  Solution. 

II.  Salt  and  Saturated  Solution. 

III.  Ice  and  Unsaturated  Solution. 

IV.  Ice  and  Cryohydrate. 
V.  Salt  and  Cryohydrate. 

shown  in  Fig.  77,  the  lettering  being  made  to  correspond  with 
that  of  the  two  diagrams.  Figs.  76  and  77,  from  which  it  is  derived. 


HETEROGENEOUS  EQUILIBRIUM  301 

(b)  Hydrated  Salt  and  Water.  An  interesting  example  is  fur- 
nished by  the  system  —  ferric  chloride  and  water.  This  system 
has  been  very  carefully  investigated  by  Roozeboom.*  The  con- 
centration-temperature diagram,  plotted  from  Roozeboom 7s  data, 


Temperature  - 
Fig.  78. 

is  given  in  Fig.  78.  The  freezing-point  of  pure  water  is  repre- 
sented by  A,  and  the  lowering  of  the  freezing-point  produced  by 
the  addition  of  ferric  chloride  is  indicated  by  the  curve  AB.  At 
the  cryohydric  temperature,  —  55°  C.,  ice,  Fe2Cl6  •  12  H2O,  sat- 
urated solution,  and  vapor  are  in  equilibrium,  and  the  system  is 
non-variant.  On  adding  more  ferric  chloride,  the  ice  phase  dis- 
appears, and  the  univariant  system,  Fe2Cl6  •  12  H2O,  saturated 
solution,  and  vapor  results.  The  equilibrium  is  represented  by  the 
curve  BC  which  may  be  regarded  as  the  solubility  curve  of  the 
dodecahydrate.  On  continuing  the  addition  of  ferric  chloride, 
the  temperature  continues  to  rise  until  the  point  C  is  reached. 
Here  the  composition  of  the  solution  is  identical  with  that  of  the 
dodecahydrate,  and,  therefore,  the  temperature  corresponding  to 
this  point,  37°  C.,  may  be  looked  upon  as  the  melting-point  of 
Fe2Cl6  •  12  H20.  Further  addition  of  ferric  chloride  will  naturally 
*  Zeit.  phys.  Chem.,  4,  31  (1889);  10,  477  (1892). 


302  THEORETICAL  CHEMISTRY 

lower  the  melting  point  and  the  equilibrium  will  alter  along  the 
curve  CD.  It  is  thus  possible  to  have  two  saturated  solutions, 
one  of  which  contains  more  water  and  the  other  less,  than  the 
hydrate  which  is  in  equilibrium  with  the  solution.  These  solu- 
tions are  both  stable  throughout  and  are  nowhere  supersaturated. 
Roozeboom  was  the  first  investigator  to  discover  a  saturated 
solution  containing  less  water  than  the  solid  hydrate  with  which 
it  is  in  equilibrium.  This  discovery  led  him  to  define  supersatu- 
ration  as  follows:  —  "A  solution  is  supersaturated  with  respect 
to  a  solid  phase  at  a  given  temperature  if  its  composition  is  between 
that  of  the  solid  phase  and  the  saturated  solution. "  At  the  point 
D  the  curve  reaches  another  minimum  which  is  analogous  to  the 
point  B,  except  that  the  heptahydrate,  Fe2Cl6  •  7  H20,  takes  the 
place  of  ice.  Here  we  have  equilibrium  between  the  dodecahydrate, 
the  heptahydrate,  saturated  solution,  and  vapor,  and  the  system 
is  non-variant.  On  further  addition  of  ferric  chloride  another 
maximum  is  reached  at  E,  corresponding  to  the  melting-point  of 
the  heptahydrate.  In  a  similar  manner,  two  other  maxima  at 
greater  concentrations  of  ferric  chloride  reveal  the  existence  of  the 
hydrates,  Fe2Cl6  •  5  H2O,  and  Fe2Cl6  -  4  H20. 

At  the  three  remaining  quadruple  points  the  following  phases  are 
in  equilibrium: — At  F,  Fe2Cl6-7  H20,  Fe2Cl6'5  H2O,  saturated  solu- 
tion and  vapor;  at  H,  Fe2Cl6  •  5  H20,  Fe2Cl6  •  4  H20,  saturated 
solution  and  vapor;  and  at  K,  Fe2Cle  •  4  H20,  Fe2Cle,  saturated 
solution  and  vapor.  The  solubility  of  the  anhydrous  salt  is 
represented  by  the  curve  KL.  Metastable  solubility  and  melting- 
point  curves  are  represented  by  dotted  lines. 

The  student  should  apply  the  phase  rule  to  this  system.  If  a 
fairly  dilute  solution  of  ferric  chloride  is  evaporated  at  31°  C.;  the 
water  gradually  disappears  and  a  residue  of  the  dodecahydrate 
remains.  This  residue  then  liquefies  and  (again  dries  down,  the 
composition  of  the  residue  corresponding  to  the  heptahydrate: 
on  further  standing  the  phenomenon  is  repeated,  the  final  and 
permanent  residue  having  a  composition  corresponding  to  the 
pentahydrate.  The  dotted  line  ab  shows  the  isothermal  along 
which  the  composition  varies.  It  would  have  been  a  difficult 
matter  to  explain  the  alternations  of  moisture  and  dryness  ob- 


HETEROGENEOUS  EQUILIBRIUM  303 

served  in  this  experiment  without  the  concentration-temperature 
diagram. 

Alloys.  Among  the  most  interesting  two-component  systems 
known  are  those  involving  mixtures  of  metals,  or  alloys.  These 
have  been  made  the  subject  of  systematic  investigations  by  num- 
erous experimenters  among  whom  may  be  mentioned  Roberts- 
Austen,  Charpy,  Roozeboom,  and  Heycock  and  Neville.  We  have 
space  to  consider  only  two  comparatively-simple  cases. 

(a)  Alloys  of  Silver  and  Copper.  The  conditions  of  equilibrium 
in  this  binary  system  have  been  studied  by  Heycock  and  Neville.* 
The  two  components,  silver  and  copper,  are  not  miscible  in  the 
solid  state  and  do  not  combine  chemically.  To  determine  the 
curves  of  equilibrium,  mixtures  of  the  two  metals  in  varying  pro- 
portions were  fused  and  then  allowed  to  cool  slowly,  the  rate  of 
cooling  being  observed  with  a  thermocouple,  one  junction  of  which 
was  maintained  at  constant  temperature,  while  the  other  junction 
was  placed  in  the  mixture  of  molten  metals.  The  terminals  of 
the  thermocouple  were  connected  to  a  sensitive  galvanometer 
graduated  to  read  directly  in  degrees,  and  the  rate  of  cooling 
was  followed  by  the  movement  of  the  needle  of  the  galvanometer. 
As  the  mixture  cooled,  two  "breaks"  were  observed;  the  first  of 
these  varied  with  the  composition  of  the  mixture,  while  the  second 
remained  practically  constant  at  777°  C.  When  the  temperatures 
corresponding  to  the  first  break  are  plotted  as  ordinates  against 
the  composition  of  the  mixture  as  abscissae,  the  diagram  shown  in 
Fig.  79  is  obtained. 

The  point  A  represents  the  freezing-point  of  pure  silver,  B  that 
of  pure  copper,  the  curve  AO  represents  the  effect  of  the  gradual 
addition  of  copper  upon  the  freezing-point  of  silver,  and  BO  the 
effect  of  silver  on  the  freezing-point  of  copper.  The  intersection 
of  the  two  curves  at  0  corresponds  to  an  alloy  containing  40  atomic 
per  cent  of  copper.  This  lowest  melting  mixture  is  known  as 
the  eutectic  («v  =  well,  and  T-^KUV  =  melt)  mixture.  At  0  the 
system  is  non-variant,  silver,  copper,  solution  and  vapor  being  in 
equilibrium.  The  solid  which  separates  at  0,  having  a  more 
uniform  texture  than  that  of  all  other  mixtures  of  the  two  com- 
*  Phil.  Trans.,  189,  25  (1897). 


304 


THEORETICAL  CHEMISTRY 


ponents,  is  known  as  the  eutectic  alloy.  When  the  composition 
of  a  mixture  of  two  metals  corresponds  to  that  of  the  eutectic 
alloy,  the  two  metals  crystallize  simultaneously  in  minute  separate 


Solution 


Ag+Solution 


Cu+Solution 


Ag+ Eutectic 


Cu+Eutectic 


1081.5 


40  at.  per  cent 

Concentration 

Fig.  79. 


Cu. 


crystals.  When  examined  under  the  microscope  the  solid  eutectic 
alloy  will  be  seen  to  be  a  conglomerate  of  very  small  crystals, 
whereas  all  of  the  other  alloys  of  the  same  metals  will  be  found  to 
contain  large  crystals  of  either  one  or  the  other  component  em- 
bedded in  the  conglomerate.  While  the  composition  of  the  eutec- 
tic alloy  in  the  above  system  is  found  to  correspond  very  closely 
to  the  formula  Ag3Cu2,  yet  the  nature  of  the  equilibrium  curves 
proves  it  to  be  nothing  more  than  a  mechanical  mixture  of  the 
two  metals.  The  meaning  of  the  diagram  will  be  clearer  from  a 
careful  consideration  of  the  phenomena  accompanying  the  cooling 
of  a  mixture  of  the  molten  metals. 

Take  for  example,  a  fused  mixture  relatively  rich  in  silver.     As 
the  temperature  falls,  a  point  will  ultimately  be  reached  at  which 


HETEROGENEOUS  EQUILIBRIUM 


305 


pure  silver  begins  to  separate,  and  since  the  temperature  remains 
constant  during  the  solidification,  a  break  occurs  in  the  cooling 
curve.  This  first  break  corresponds  to  a  point  on  the  curve  AO. 
As  silver  continues  to  separate,  the  composition  of  the  mixtures 
changes  along  AO,  until  when  0  is  reached,  the  mixture  is  satu- 
rated with  respect  to  copper,  and  both  metals  separate  as  a  con- 
glomerate having  the  same  composition  as  the  fused  mixture. 
The  separation  of  the  eutectic  alloy  causes  the  second  break  in 
the  cooling  curve,  the  temperature  remaining  constant  until  the  en- 
tire mass  has  solidified.  It  will  be  noticed  that  this  system  is  the 
exact  analogue  of  the  system — anhydrous  salt  and  water;  the  eutec- 
tic point  and  the  cryohydric  point  representing  identical  conditions, 
(b)  Alloys  of  Gold  and  Aluminium.  This  system  has  been 
studied  by  Roberts-Austen.*  The  equilibrium  curves  in  the 
concentration  temperature  diagram,  Fig.  80,  reveal  the  existence 


Au  Composition 

Fig.  80. 

*  Phil.  Trans.  A.,  194,  201  (1900). 


306  THEORETICAL  CHEMISTRY 

of  definite  compounds,  AusAl2,  Au2Al,  and  AuAl2,  corresponding 
to  the  points  D,  E,  and  H  respectively.  The  discontinuities  at  B 
and  G  suggest  the  possibility  of  two  other  compounds,  viz.,  Au4Al 
and  AuAl.  The  diagram  shows  that  the  following  substances 
will  crystallize  in  succession  from  the  molten  alloy,  these  being 
the  different  solids  with  which  the  liquid  mixture  is  saturated  in 
its  successive  stages  of  equilibrium :  — 

Curve  AB,  pure  gold  at  A ; 

Curve  BC,  Au^Al,  nearly  pure  at  J5; 

Curve  CD,  Au5Al2  or  Au8Al3,  nearly  pure  at  D; 

Curve  DEF,  Au2Al,  pure  at  E\ 

Curve  FG,  AuAl,  maximum  undetermined; 

Curve  GHI,  AuAl2,  pure  at  H', 

Curve  IJ,  Al,  pure  at  J. 

The  points  (7,  F,  and  I  represent  non-variant  systems,  the  melt- 
ing points  of  the  respective  eutectic  alloys  being  527°,  569°,  and 
647°.  This  system  in  many  respects  resembles  the  system  —  ferric 
chloride  and  water. 

Three-component  Systems.  When  three  components  are  pres- 
ent, the  equilibria  become  much  more  complicated.  Applying 
the  formula,  C  —  P  +  2  =  F,  we  find  that  it  is  necessary  to 
have  five  phases  co-existent  for  a  non-variant  system,  four  for  a 
uni variant,  three  for  a  bivariant,  and  two  for  a  tri variant.  The 
most  satisfactory  method  of  representing  equilibria  in  three-com- 
ponent systems  is  that  in  which  use  is  made  of  the  triangular 
diagram.  The  three  corners  of  an  equilateral  triangle  are  taken 
to  represent  the  pure  components,  and  the  composition  of  any 
mixture,  expressed  in  atomic  percentages,  is  represented  by  the 
position  of  the  center  of  mass  of  the  three  components  within 
the  triangle. 

For  example,  in  the  system,  —  potassium  nitrate,  sodium  nitrate, 
and  lead  nitrate,  carefully  investigated  by  Guthrie,*  the  three 
components  are  placed  at  the  corners  of  the  triangle  shown  in 

*  Phil.  Mag.,  5,  17,  472  (1884). 


HETEROGENEOUS  EQUILIBRIUM  307 

Fig.  81.    The  melting-point  of  pure  potassium  nitrate  is  340° 
and  that  of  pure  sodium  nitrate  is  305°.     The  melting-point  of 


IJa  N0« 

340°  215° 

Fig.  81. 

pure  lead  nitrate  cannot  be  determined  since  the  salt  decomposes 
before  its  melting-point  is  reached.  The  eutectic  mixtures  of 
the  three  pairs  of  salts  are  represented  by  the  points  D,  E,  and  F 
respectively.  In  like  manner  0  represents  the  melting  point  of 
the  non-variant  system,  —  potassium  nitrate,  sodium  nitrate,  lead 
nitrate,  fused  mixture  of  the  three  salts,  and  vapor.  In  order 
to  represent  temperature,  use  is  frequently  made  of  a  triangular 
prism  in  which  the  altitude  is  taken  as  the  temperature  axis,  the 
resulting  surface  within  the  prism  representing  the  variation  of 
the  equilibrium  with  temperature.* 

PROBLEMS. 

1.  The  vapor  pressure  of  solid  NH4HS  at  25°.l  is  50.1  cm.  Assuming 
that  the  vapor  is  practically  completely  dissociated  into  NH3  and  H2S, 
calculate  the  total  pressure  at  equilibrium  when  solid  NH4HS  is  allowed 

*  For  a  complete  treatment  of  three-component  systems  as  well  as  for  a 
clear  presentation  of  the  phase  rule,  the  student  should  consult  "  The  Phase 
Rule  and  Its  Applications,"  by  Alexander  Findlay. 


308  THEORETICAL  CHEMISTRY 

to  dissociate  at  25°.  1  in  a  vessel  containing  ammonia  at  a  pressure  of 
32  cm.  Ans.  59.5  cm. 

2.  In  the  partition  of  acetic  acid  between  CC14  and  water,   the  con- 
centration of  the  acetic  acid  in  the  CC14  layer  was  c  gram-molecules  per 
liter  and  in  the  corresponding  water  layer  w  gram-molecules  per  liter. 

c  0.292  0.363  0.725  1.07  1.41 

w         4.87  5.42  7.98  9.69  1.07 

Acetic  acid  has  its  normal  molecular  weight  in  aqueous  solutions.  From 
these  figures  show  that,  at  these  concentrations,  the  acetic  acid  in  the 
carbon  tetrachloride  solution  exists  as  double  molecules. 

3.  Acetic  acid  distributes  itself  between  water  and  benzene  in  such  a 
manner  that  in  a  definite  volume  of  water  there  are  0.245  and  0.314  gram 
of  the  acid,  while  in  an  equal  volume  of  benzene  there  are  0.043  and 
0.071  gram.    What  is  the  molecular  weight  of  acetic  acid  in  benzene, 
assuming  it  to  be  normal  in  water?  Ans.   122.2. 

4.  The  salt  Na2HP04.12  H20  has  a  vapor  pressure  at  15°  of  8.84  mm., 
and  at  17°.3  of  10.53  mm.     Calculate  the  heat  of  vaporization,  i.e.,  the 
thermal  change  during  the  loss  of  1  mol  of  water  of  crystallization  by 
evaporation.  Ans.    — 12,728  cal. 

5.  The  solubility  of  boric  acid  in  water  is  38.45  grams  per  liter  at  13°, 
and  49.09  grams  per  liter  at  20°.    Calculate  the  heat  of  solution  of  boric 
acid  per  mol.  Ans.    —5840  cal. 

6.  Plot  the  pressure-temperature  diagram  for  calcium  carbonate  from 
the  table  given  on  p.  280,  and  apply  the  phase  rule. 

7.  Is  it  possible  to  decide  by  the  phase  rule  whether  the  eutectic  alloy 
is  a  mixture  or  a  compound? 


CHAPTER  XIV. 
CHEMICAL  KINETICS. 

Velocity  of  Reaction.  In  the  two  preceding  chapters  we  have 
considered  the  equilibrium  which  is  established  when  the  speeds 
of  the  direct  and  reverse  reactions  have  become  equal.  We  now 
proceed  to  consider  the  velocity  of  individual  reactions.  By  far 
the  greater  number  of  the  reactions  between  inorganic  substances 
proceed  with  such  rapidity  that  it  is  impossible  to  measure  their 
velocities.  Thus,  when  an  acid  is  neutralized  by  a  base,  the  indi- 
cator changes  color  almost  instantly.  There  are  a  few  well- 
known  reactions  which  are  exceptions  to  this  rule;  among  these 
may  be  mentioned  the  oxidation  of  sulphur  dioxide  and  the  de- 
composition of  hydrogen  peroxide.  Both  of  these  reactions  are 
well  adapted  to  kinetic  experiments.  In  organic  chemistry,  on 
the  other  hand,  slow  reactions  are  the  rule  rather  than  the  excep- 
tion. Thus  the  reaction  between  an  alcohol  and  an  acid  forming 
an  ester  and  water,  proceeds  very  slowly  under  ordinary  condi- 
tions and  the  progress  of  the  reaction  may  be  easily  followed. 
By  means  of  the  law  of  mass  action  it  is  possible  to  derive  equations 
expressing  the  velocity  of  a  reaction  at  any  moment  in  terms  of 
the  concentrations  of  the  reacting  substances  present  at  that  time. 

Let  the  equation 


represent  a  reversible  reaction  and  let  a,  6,  c,  and  d  be  the  respec- 
tive initial  concentrations  of  the  reacting  substances  AI,  A2)  AI, 
and  A*.  The  velocity  of  the  direct  reaction  will  then  be 

dx 

^  =  k  (a  -  x)  (b-  x),  (1) 

where  k  is  the  velocity  constant,  and  ax  is  the  infinitely  small 
increase  in  the  amount  of  x  during  the  infinitely  small  interval 

309 


310  THEORETICAL  CHEMISTRY 

of  time  t.  Similarly  the  velocity  of  the  reverse  reaction  will 
be 

^  =  A?i(c  +  x)  (d  +  x).  (2) 

It  is  evident  that  the  substances  on  the  right-hand  side  of  the  equa- 
tion will  exert  an  ever-increasing  influence  upon  the  velocity  of 
the  direct  reaction,  which  must  accordingly  decrease.  When, 
however,  the  velocities  of  the  direct  and  reverse  reactions  become 
equal,  equilibrium  will  be  established,  and  the  ratio  of  the  amounts 
of  the  reacting  substances  on  the  two  sides  of  the  equation  will 
remain  constant.  The  total  velocity  due  to  these  opposing  reac- 
tions will  be 

--        =  *(«-*)  (&  -*)  -k  («  +  *)(*  +  *)     (3) 


and  at  equilibrium,  when  -57-  =  0, 

k  (a  —  x)  (b  —  x)  =  ki  (c  -f  x)  (d  +  #), 
or 

(c  +  x)  (d  +  x)       k 
(a  -x)(b-  x)      ki 

This  equation  has  been  thoroughly  tested  in  the  two  preceding 
chapters.  Thus,  in  the  reaction 

C2H5OH  +  CH3COOH<=»  CH3COOC2H5  +  H20, 

Kc  has  been  shown  to  have  the  value,  2.84,  at  ordinary  temper- 
atures. The  velocity  constants  of  the  direct  and  reverse  reactions 
have  also  been  determined,  the  values  being,  k  =  0.000238  and 
ki  =  0.000815.  When  these  values  are  substituted  in  the  equa- 

k 

tion,   j-  =  Kc,  we  obtain  Kc  =  2.92,  a  value  which  agrees  well 
KI 

with  that  found  by  direct  experiment.  The  application  of  equa- 
tion (3)  is  much  simplified  by  the  fact  that  most  reactions  proceed 
nearly  to  completion  in  one  direction,  so  that  the  term  ki  (c  +  x) 
(d  +  x)  will  be  so  small  that  it  may  be  neglected.  We  then  have 

=  k  (a  -  x)  (b  -  x),  (5) 


CHEMICAL  KINETICS 


311 


an  equation  expressing  the  velocity  of  the  direct  reaction  in  terms 
of  the  concentrations  of  the  reacting  substances. 

Unimolecular  Reactions.  The  simplest  type  of  chemical 
reaction  is  that  in  which  only  one  substance  undergoes  change 
and  in  which  the  velocity  of  the  reverse  reaction  is  negligible.  The 
decomposition  of  hydrogen  peroxide  is  an  example  of  such  a  reac- 
tion. In  the  presence  of  a  catalyst  such  as  certain  unorganized 
ferments  or  collodial  platinum,  hydrogen  peroxide  decomposes 
as  represented  by  the  equation 


This  reaction  is  usually  allowed  to  take  place  in  dilute  aqueous 
solution  so  that  there  is  no  appreciable  alteration  in  the  amount 
of  solvent  throughout  the  entire  course  of  the  reaction.  Further- 
more, the  activity  of  the  catalyst  remains  constant  so  that  the 
course  of  the  reaction  is  wholly  determined  by  the  concentration 
of  the  hydrogen  peroxide.  A  very  satisfactory  catalyst  is  haBmase, 
an  enzyme  derived  from  blood.  The  concentration  of  hydrogen 
peroxide  present  at  any  time  during  the  reaction  can  be  deter- 
mined very  simply  by  removing  a  definite  portion  of  the  reaction 
mixture,  adding  an  excess  of  sulphuric  acid  to  destroy  the  activity 
of  the  hsemase,  and  then  titrating  with  a  standard  solution  of 
potassium  permanganate. 

The  following  table  gives  the  results  of  such  an  experiment:  — 


t  (minutes). 

cc.  KMnO4. 

X, 

cc.  KMnO4. 

k 

0 

46.1 

0 

5 

37.1 

9.0 

0.0435 

10 

29.8 

16.3 

0.0438 

20 

19.6 

26.5 

0.0429 

30 

12.3 

33.8 

0.0440 

50 

5.0 

41.1 

0.0444 

Mean    0.0437 

The  second  column  of  the  table  gives  the  number  of  cubic 
centimeters  of  the  potassium  permanganate  solution  required  to 
oxidize  25  cc.  of  the  reaction  mixture  when  the  time  intervals 


312  THEORETICAL  CHEMISTRY 

recorded  in  the  first  column  have  elapsed  after  the  introduction 
of  the  catalyst.  Since  the  numbers  in  the  second  column  repre- 
sent the  actual  concentration  of  hydrogen  peroxide  present  at 
the  end  of  the  successive  intervals  of  time,  it  is  evident  that  the 
difference  between  these  numbers  and  46.1  cc.  —  the  initial 
concentration  of  hydrogen  peroxide  —  will  give  the  amounts  of 
peroxide  decomposed  in  those  intervals.  These  numbers  are 
recorded  in  the  third  column  of  the  table.  It  will  be  seen  that 
as  the  concentration  of  the  hydrogen  peroxide  decreases  the  rate 
of  the  reaction  diminishes.  Thus,  in  the  first  interval  of  10  min- 
utes, an  amount  of  hydrogen  peroxide  corresponding  to  46.1  — 
29.8  =  16.3  cc.  of  potassium  permanganate  is  decomposed,  while 
in  the  second  interval  of  10  minutes,  the  amount  of  hydrogen 
peroxide  decomposed  is  equivalent  to  29.8  —  19.6  =  10.2  cc.  of 
potassium  permanganate.  Since  only  a  single  substance  is  under- 
going change,  equation  (5)  simplifies  to  the  following  form:  — 

dx  . 


It  is  impossible  to  apply  the  equation  in  this  form,  since  in  order 
to  obtain  accurate  titrations,  dt  must  be  taken  fairly  large  and 
during  this  interval  of  time  a  —  x  would  have  diminished.  Approx- 
imate values  of  k  may  be  obtained  by  taking  the  average  value 
of  a  —  x  during  the  interval  dt  within  which  an  amount  dx  of  hydro- 
gen peroxide  is  being  decomposed.  For  example,  let  us  take  the 
interval  between  5  and  10  minutes;  dx  =  16.3  —  9.0  =  7.3  cc., 
dt  =  5  min.,  and  the  average  value  of  a  —  x  is 

37.1  +  19.8      QQA_ 

jr =  33.45  cc. 

& 

Substituting  in  the  equation 

—    •     5 

dt 
we  have 

^  =  k  X  33.45, 

and 

k  =  0.0436. 


CHEMICAL  KINETICS  313 

Similarly  taking  the  next  interval  between  10  and  20  minutes; 
dx  =  26.5  —  16.3  =  10.2  cc.,  dt  =  10  minutes,  and  the  average 

value  of  a  —  x  is  —  '-  —  ~  -  —  =  24.7   cc.     Substituting   in  the 

a 

equation  as  before,  we  obtain 

and 

k  =  0.0413. 

As  will  be  seen  these  two  values  of  k  are  not  in  good  agreement, 
although  the  first  value  of  k  agrees  closely  with  the  mean  value 
of  k  given  in  the  fourth  column  of  the  table. 
In  order  to  apply  the  equation 

dx      ,  f          ^. 
Tt=k(a-x) 

it  must  be  integrated.* 

The  integration  of  this  equation  may  be  performed  as  follows:  — 


dx      1  ( 
-  =  *  (a  - 


therefore 


a  —  x 
Integrating,  we  have 

/j  /* 

--    I  k  dt  =  constant  =  C, 
a  —  x      J 

therefore 

-  log*  (a  -  x)  -  kt  =  C. 

In  order  to  determine  C,  the  constant  of  integration,  we  make 
use  of  the  experimental  fact  that  when  t  =  0,  x  =  0.  Substitut- 
ing these  values,  we  have 

-  loge  a  =  C. 
Consequently 

loge  a  —  loge  (a  —  x)  =  kt, 
or 


*  The  student  who  is  unfamiliar  with  the  Calculus  must  take  the  result; 
Of  this  calculation  for  granted, 


314  THEORETICAL  CHEMISTRY 

Passing  to  Briggsian  logarithms,  we  obtain 

ilog— ^—  =  0.4343  k. 

t      &a  —  x 

By  substituting  in  this  equation  the  corresponding  values  of  a, 
a  —  x}  and  t  from  the  preceding  table,  the  values  of  k  given  in  the 
fourth  column  of  the  table  are  obtained. 
The  equation 

dx      ,  ,          N 
-  =  k(a-x), 

may  also  be  thrown  into  an  exponential  form,  as  follows :  — 
Since 

a 


t      3  a  -  x 
we  may  write, 

,,      .      a  —  x 
—  kt  =  loge •> 

or 

a  —  x  =  ae~kty 
and 

x  =  a  (1  -  e~kt). 

In  this  equation  k  may  be  regarded  as  the  fraction  of  the  total 
amount  of  substance  decomposing  in  the  unit  of  time,  provided 
this  unit  is  so  small  that  the  quantity  at  the  end  of  the  time  unit 
is  only  slightly  different  from  that  at  the  beginning.  The  time 
required  for  one-half  of  the  substance  to  change,  is  known  as  the 
period  of  half-change,  T,  and  may  be  calculated  from  k  by  means 
of  the  equation 

log  0.5  =  0.4343  fc!T, 
therefore 


or 


T  =  0.6943 
K 


=  1.443  T. 
k 


Reactions  in  which  only  one  mol  of  a  single  substance  undergoes 
change  are  known  as  unimolecular  reactions,  or  reactions  of  the 


CHEMICAL  KINETICS  315 

first  order.  In  a  unimolecular  reaction,  the  velocity  constant  k, 
is  independent  of  the  units  in  which  concentration  is  expressed. 
If,  in  the  integrated  equation 

kt  =  \oge » 

a  —  x 

t  becomes  infinite,  then  x  =  a.  In  other  words,  for  finite  values 
of  t}  x  must  always  remain  less  than  a  and  the  reaction  will  never 
proceed  to  completion. 

Another  unimolecular  reaction  which  has  been  thoroughly 
investigated,  is  the  hydrolysis  of  cane  sugar.  When  cane  sugar 
is  dissolved  in  water  containing  a  small  amount  of  free  acid  it  is 
slowly  transformed  into  d-glucose  and  d-fructose.  The  velocity 
of  the  reaction  is  very  small  and  is  dependent  upon  the  strength 
of  the  acid  added.  The  progress  of  the  reaction  may  be  very  easily 
followed  by  means  of  the  polarimeter.  Cane  sugar  itself  is  dex- 
tro-rotatory, while  d-fructose  rotates  the  plane  of  polarization 
more  strongly  to  the  left  than  d-glucose  rotates  it  to  the  right. 
Therefore,  as  the  hydrolysis  proceeds,  the  angle  of  rotation  to  the 
right  steadily  diminishes  until,  when  the  reaction  is  complete,  the 
plane  of  polarization  will  be  found  to  be  rotated  to  the  left.  On 
this  account  the  hydrolysis  of  cane  sugar  is  commonly  termed 
inversion  and  the  molecular  mixture  of  d-fructose  and  d-glucose 
constituting  the  product  of  the  reaction  is  called  invert  sugar. 
Let  «o  denote  the  initial  angle  of  rotation,  at  the  time  t  =  0, 
due  to  a  mols  of  cane  sugar,  let  <*</  denote  the  angle  of  rotation 
when  inversion  is  complete  and  let  a  be  the  angle  of  rotation  at 
any  time  t',  then  since  rotation  of  the  plane  of  polarization  is  pro- 
portional to  the  concentration  x,  the  amount  of  cane  sugar  in- 
verted, will  be 

do  —  a 

x  =  a — — — 7- 
<*o  +  «o 

In  the  equation 

Ci2H22Oii  +  H20  «=±  C6Hi206  +C6Hi206, 

representing  the  inversion  of  cane  sugar,  the  velocity  of  the 
reaction  will  be,  according  to  the  law  of  mass  action,  proportional 
to  the  molecular  concentrations  of  the  cane  sugar  and  the  water. 


316 


THEORETICAL  CHEMISTRY 


Since  the  reaction  takes  place  in  the  presence  of  such  a  large  excess 
of  water,  its  effect  may  be  considered  to  be  constant.  The 
velocity  of  the  reaction  is  then  proportional  to  the  active  mass 
of  the  sugar  alone,  or  in  other  words  the  reaction  is  unimolecular. 
In  the  differential  equation  expressing  the  velocity  of  a  unimolec- 
ular reaction, 

-j7  =  k  (a  —  x)  t 

we  have 

a 


7      1 
k  =  - 

t 


a  —  x 


and  since  a  and  x  are  measured  in  terms  of  angles  of  rotation  of 
the  plane  of  polarization,  we  have 


t          a  -f-  ao 

The  following  table  gives  the  results  obtained  with  a  20  per  cent 
solution  of  cane  sugar  in  the  presence  of  0.5  molar  solution  of  lactic 
acid  at  25°  C. 


t  (minuted). 

a 

1 

o 

34°.  5 

1,435 

31°.  1 

0.2348 

4,315 

25°.  0 

0.2359 

7,070 

20°.  16 

0.2343 

11,360 

13°.  98 

0.2310 

14,170 

10°.  01 

0.2301 

16,935 

7°.  57 

0.2316 

19,815 

5°.  08 

0.2991 

29,925 

-  1°.65 

0.2330 

Inf. 

-10°.  77 

Bimolecular  Reactions.  When  two  substances  react  and  the 
concentration  of  each  changes,  the  reaction  is  bimolecular  or  of 
the  second  order.  Let  a  and  b  represent  the  initial  molar  con- 
centrations of  the  two  reacting  substances  and  let  x  denote  the 
amount  transformed  in  the  interval  of  time  t',  then  the  velocity 
of  the  reaction  will  be  expressed  by  the  equation 

dx 

-    =  k  (a  -  x)  (b  -  x). 


CHEMICAL  KINETICS  317 

The  simplest  case  is  that  in  which  the  two  substances  are  present 
in  equivalent  amounts.  Under  these  conditions  the  velocity 
equation  becomes 


This  equation  may  be  integrated  as  follows:  — 


therefore 

z.  a      i\      [    1    I3"3  afc-si 

K  (k  —  1  1  )  =1  -  I      =  -,  -  r-7 

[a  -  x]Xl      (a  -  xi)  (a  - 


or 


If  time  be  reckoned  from  the  beginning  of  the  reaction,  then  Xi  =  0 
and  t  =  0,  and  we  have 

7       1  x 

k  =  -  - 


t     a  (a  —  x)  ' 

If  the  reacting  substances  are  not  present  in  equivalent  amounts 
then  the  velocity  equation  becomes 

dx      T  f          N  ,,         >. 
_  =  k  (a  -  x)  (b  -  x). 

Assuming  that  time  is  measured  from  the  beginning  of  the  reaction, 
the  integration  of  this  equation  may  be  performed  as  follows:  — 

dx 


'o  Jo    (a  —  x)  (b  —  x) 

Decomposing  into  partial  fractions, 


a  —  b  |_ Jo  b  —  x 

*  The  student  who  is  unfamiliar  with  the  Calculus  must  take  the  results 
of  these  calculations  for  granted. 


318  THEORETICAL  CHEMISTRY 

therefore 


or 

n         1     i       b  (a  —  x) 

kt  =  -  rloge  —  77  -  ? 

a-b     =e  a  (b  -  x) 
Or  passing  to  Briggsian  logarithms, 

0.4343  fc  =  -l        lob(" 


The  value  of  k  in  a  bimolecular  reaction  is  not  independent  of 
the  units  in  which  the  concentration  is  expressed,  as  is  the  case 
with  a  unimolecular  reaction.  Suppose  that  a  unit  I/nth  of 
that  originally  selected  is  used  to  express  concentration,  then  the 
value  of  k  in  the  equation 


t  a(a  —  x) ' 
becomes 

, ,  _  1  nx  1 


t    no,  •  n  (a  —  x)      t    na(a  —  x) 

Thus,  the  value  of  k  varies  inversely  as  the  numbers  expressing 
the  concentrations. 

As  an  illustration  of  a  bimolecular  reaction  we  may  take  the 
hydrolysis  of  an  ester  by  an  alkali.     The  reaction 

CH3COOC2H5  +  NaOH  <=±  CH3COONa  +  C2H6OH, 

has  been  studied  by  Warder,*  Reicher,f  Arrhenius, }  Ostwald  § 
and  others.  Arrhenius  employed  in  his  experiments  0.02  molar 
solutions  of  ester  and  alkali.  These  solutions  were  placed  in 
separate  flasks  and  warmed  to  25°  C.  in  a  thermostat  maintained 
at  that  temperature;  equal  volumes  were  then  mixed,  and  at 
frequent  intervals  a  portion  of  the  reaction  mixture  was  removed 

*  Berichte,  14,  1361  (1881). 

t  Lieb.  Ann.,  228,  257  (1885). 

J  Zeit.  phys.  Chem.,  i,  110  (1887). 

§  Jour,  prakt.  Chem.,  35,  112  (1887). 


CHEMICAL  KINETICS 


319 


and  titrated  rapidly  with   standard   acid.     The 
table  contains  some  of  the  results  obtained :  — 


accompanying 


t  (minutes). 

a  —  x 

k 

0 

8.04 

4 

5.30 

0.0160 

6 

4.58 

0.0156 

8 

3.91 

0.0164 

10 

3.51 

0.0160 

12 

3.12 

0.0162 

Mean  0.0160 

The  numbers  in  the  second  column  of  the  table  represent  the 
concentrations  of  sodium  hydroxide  and  of  ethyl  acetate,  expressed 
in  terms  of  the  number  of  cubic  centimeters  of  standard  acid 
required  to  neutralize  10  cc.  of  the  reaction  mixture.  Owing  to 
the  high  velocity  of  the  reaction  it  is  difficult  to  avoid  large  experi- 
mental errors,  nevertheless  the  values  of  k  given  in  the  third  column 
of  the  table  will  be  observed  to  differ  very  slightly  from  the  mean 
value. 

Reicher  investigated  the  same  reaction  when  the  reacting  sub- 
stances were  not  present  in  equivalent  proportions.  In  this  case, 
the  progress  of  the  reaction  was  followed  by  titrating  definite 
portions  of  the  reaction  mixture  from  time  to  time,  the  excess  of 
sodium  hydroxide  being  determined  by  titrating  a  portion  of 
the  mixture  at  the  expiration  of  twenty-four  hours,  when  the 
ester  was  completely  hydrolyzed.  His  results  are  given  in  the 
following  table:  — 


t  (minutes). 

a—x 
(alkali). 

b-x 
(ester). 

i 

0 
4.89 
11.36 
29.18 
Inf. 

61.95 
50.59 
42.40 
29.35 
14.92 

47.03 
35.67 
27.48 
14.43 
0 

6!  00093 
0.00094 
0.00092 

320  THEORETICAL  CHEMISTRY 

Reicher  also  studied  the  effect  of  different  bases  upon  the  ve- 
locity of  the  reaction.  He  found  for  strong  bases  approximately 
equal  values  of  k}  but  for  weak  bases  the  values  were  irregular 
and  smaller  than  those  obtained  with  the  more  completely  ionized 
bases.  Arrhenius  pointed  out  that  the  hydrolyzing  power  of  a 
base  is  proportional  to  the  number  of  hydroxyl  ions  which  it 
yields.  Writing  the  equation  for  the  above  hydrolysis  in  terms  of 
ions,  we  have 

CH3COOC2H5  +  Na'  +  OH'  <=>  CH3COO'  +  Na"  +  C2H5OH. 

It  is  evident  from  this  equation  that  all  bases  furnishing  the  same 
number  of  hydroxyl  ions  should  give  identical  values  of  k.  We 
may,  therefore,  modify  the  fundamental  differential  equation  as 

follows:  — 

tff 

~  =  k'a  (a  -x)(b-  x), 

where  a  is  the  degree  of  ionization  of  the  base. 

Trimolecular  Reactions.  When  equivalent  quantities  of  three 
substances  react,  the  reaction  is  trimolecular  or  of  the  third  order. 
If  the  initial  molar  concentrations  of  the  reacting  substances  are 
denoted  by  a,  6,  and  c,  and  if  x  denotes  the  proportion  of  each 
which  is  transformed  in  the  interval  of  time  t,  the  velocity  of  the 
reaction  will  be  represented  by  the  differential  equation 

-rr  =  k  (a  —  x)  (b  —  x}  (c  —  v). 

If  the  substances  are  present  in  equivalent  amounts,  the  equation 
becomes 

~-j7  =  k  (a  —  x)3, 

an  expression  which  is  much  less  difficult  to  integrate. 

The  integration  of  this  equation  may  be  performed  as  follows.* 


(\o  ' 
a-xY 

*  The  student  who  is  unfamiliar  with  the  Calculus  must  take  the  results 
of  these  calculations  for  granted. 


CHEMICAL  KINETICS  321 

therefore 

kt 


hence 


xr  i 

2L(a-x)2J0' 

ir   *    .IT 

2[>-z)2      a2J 


JL  =  1  .  x(2a—  x) 


When  the  reacting  substances  are  not  taken  in  equivalent  amounts, 
the  integration  of  the  velocity  equation  may  be  performed  as 
follows :  — 

rl*r 

~  =  k(a-x)(b-x)(c-  x), 
therefore 

If  fit  —  ^x 

(a  —  x)  (b  —  x)  (c  —  x)  * 

Decomposing  into  partial  fractions, 

A  *          C 


(a  —  x)  (b  —  x)  (c  —  x)      a  —  x      b  —  x      c  —  x 
Multiplying  through  by  (a  —  x),  we  obtain 

1  ,          ,  (     B  C 

(b-x)(c-x)  =  A  +  (a  -  *}  J  b^x  +  t= 

Let  x  =  a,  then 


(a  -b)(c-  a) 

Similarly,  multiplying  by  (b  —  x)  and  (c  —  x),  and  then  placing 
x  =  b,  and  x  =  c,  we  have 

» 1 

(a -6)  (b-c)' 
and 


(6  -  c)  (c  -  a) 
Then  we  obtain  by  substitution 


c* <fof =  _         i          r*  dx 

Jo   (a  —  x)  (b  —  x)  (c  —  x)          (a  —  b)  (c  —  a)JQ  a  —  x 

1  C*    dx 1  C*    da 

(a  —  6)  (6  —  c)  Jo   b  —  x      (b  —  c)  (c  —  a)JQ  c  — 


dx_ 
x 


THEORETICAL  CHEMISTRY 


322 

Therefore, 

=  [-  («  -  6HC  -  a) log«  al  ~  [(a  -  b)\b  -  c)  log<  6]J 

-[(fc-cAc-^Ho' 


kt 


or 


t  (a  -  6)  (6  -  c)  (c  -  a) 

In  a  trimolecular  reaction,  fc  is  inversely  proportional  to  the  square 
of  the  original  concentration. 

A  typical  trimolecular  reaction  is  that  between  ferric  and 
stannous  chlorides.  This  reaction,  represented  by  the  following 
equation 

2  FeCl3  +  SnCl2  <=±  2  FeCl2  +  SnCl4, 

has  been  investigated  by  A.  A.  Noyes.*  Dilute  solutions  of  the 
reacting  substances  were  mixed  at  constant  temperature,  and 
definite  portions  of  the  reaction  mixture  were  removed  at  meas- 
ured intervals  of  time  and  titrated  for  ferrous  iron.  Before 
titrating  with  a  standard  solution  of  potassium  permanganate  it 
was  necessary  to  decompose  the  stannous  chloride  present  with 
mercuric  chloride.  The  following  table  gives  the  results  obtained 
with  0.025  molar  solutions  of  ferric  chloride  and  stannous  chloride. 


t  (minutes). 

a—  x 

X 

k 

2.5 

0.02149 

0.00351 

113 

3 

0.02112 

0.00388 

107 

6 

0.01837 

0.  00663  x 

114 

11 

0.01554 

0.00946 

116 

15 

0.01394 

0.01106 

118 

18 

0.01313 

0.01187 

117 

30 

0.01060 

0.01440 

122 

60 

0.00784 

0.01716 

122 

Mean  116 

Noyes  also  found  that  the  velocity  of  the  reaction  is  accelerated 
more  by  an  excess  of  ferric  chloride  than  by  an  equal  excess  of 
stannous  chloride. 

*  Zeit.  phys.  Chem.,  16,  546  (1895). 


CHEMICAL  KINETICS  323 

Reactions  of  Higher  Orders.  Reactions  of  the  fourth,  fifth 
and  eighth  orders  have  recently  been  investigated,  but  examples 
of  reactions  of  orders  higher  than  the  third  are  extremely  rare. 
This  fact  is  at  first  sight  surprising  since  the  equations  of  many 
chemical  reactions  involve  a  large  number  of  molecules,  and  we 
would  naturally  expect  the  order  of  such  reactions  to  be  corre- 
spondingly high.  For  example,  the  reaction  represented  by  the 

equation,  2  PH3  +  4  02  =  P205  +  3  H20, 

involves  six  molecules  of  the  substances  initially  present  and, 
therefore,  we  should  infer  it  to  be  a  reaction  of  the  sixth  order. 

Kinetic  experiments  by  van  der  Stadt  have  shown  it  to  be  a 
bimolecular  reaction,  the  velocity  of  reaction  being  proportional 
to  the  concentrations  of  the  phosphine  and  the  oxygen.  On 
allowing  the  gases  to  mix  slowly  by  diffusion,  it  was  discovered 
that  the  reaction  actually  takes  place  in  several  successive  stages, 
the  first  stage  being  represented  by  the  equation  of  the  bimolec- 
ular  reaction  PH3  +  02  =  HPO2  +  H2. 

The  subsequent  changes  involving  the  oxidation  of  the  products 
of  this  reaction  take  place  with  great  rapidity.  It  is  highly 
probable  that  the  equations  which  are  ordinarily  employed  to 
represent  chemical  reactions  really  represent  only  the  initial  and 
final  stages  of  a  series  of  relatively  simple  reactions.  Larmor* 
has  shown  that  when  chemical  reactions  are  considered  from  the 
molecular  standpoint,  the  bimolecular  reaction  is  the  most  prob- 
able. He  says,  "Imagine  a  substance,  say  gaseous  for  simplicity, 
formed  by  the  immediate  spontaneous  combination  of  three  gas- 
eous components  A,  B,  and  C.  When  these  gases  are  mixed,  the 
chances  are  very  remote  of  the  occurrence  of  the  simultaneous 
triple  encounter  of  an  A,  a  B,  and  a  C,  which  would  be  necessary 
to  the  immediate  formation  of  an  ABC;  whereas  if  ever  formed, 
it  would  be  liable  to  the  normal  chance  of  dissociating  by  collisions; 
it  would  thus  be  practically  non-existent  in  the  statistical  sense. 
But  if  an  intermediate  combination  AB  could  exist,  very  tran- 
siently, though  long  enough  to  cover  a  considerable  fraction  of  the 
*  Proc.  Manchester  Phil.  Soc.,  1908. 


324  THEORETICAL  CHEMISTRY 

mean  free  path  of  the  molecules,  this  will  readily  be  formed  by 
ordinary  binary  encounters  of  A  and  B,  and  another  binary 
encounter  of  AB  with  C  will  now  form  the  triple  compound  ABC 
in  quantity. " 

Determination  of  the  Order  of  a  Reaction.  It  has  been 
shown  in  the  foregoing  pages  that  the  time  required  to  complete 
a  certain  fraction  of  a  reaction  is  dependent  upon  the  order  of 
the  reaction  in  the  following  manner :  — 

(1)  In  a  unimolecular  reaction  the  value  of  k  is  independent  of 
the  initial  concentration; 

(2)  In  a  bimolecular  reaction  the  value  of  k  is  inversely  pro- 
portional to  the  initial  concentration; 

(3)  In  a  trimolecular  reaction  the  value  of  k  is  inversely  pro- 
portional to  the  square  of  the  initial  concentration. 

Hence,  in  general,  in  a  reaction  of  the  nth.  order,  the  value  of 
k  is  inversely  proportional  to  the  (n  — •  1)  power  of  the  initial  con- 
centration. If  the  value  of  k  is  determined  with  definite  concen- 
trations of  the  reacting  substances,  and  then  with  multiples  of 
those  concentrations,  the  order  of  the  reaction  can  be  determined 
according  to  the  above  rules  by  observing  the  manner  in  which  k 
varies  with  the  concentration. 

The  order  of  a  reaction  may  also  be  readily  determined  by  means 
of  a  graphic  method.  Thus,  to  determine  the  order  of  a  reaction 
we  ascertain  by  actual  trial  whicli  one  of  the  following  expressions, 
in  which  C  denotes  concentration,  will  give  a  straight  line  when 
plotted  against  times  as  abscissae :  — 

(1)  log  C  —  reaction  unimolecular; 

(2)  1/C  — reaction  bimolecular; 

(3)  1/C2  —  reaction  trimolecular; 

(4)  l/Cn  —  reaction  n  +  1  molecular. 

Complex  Reaction  Velocities.  Thus  far  we  have  considered 
the  velocity  of  reactions  which  are  practically  complete.  There 
are  numerous  cases,  however,  in  which  the  course  of  the  reaction 
is  complicated  by  such  disturbing  factors  as  (1)  counter  reactions, 
(2)  side  reactions,  and  (3)  consecutive  reactions.  These  disturb- 
ing causes  will  now  be  considered. 


CHEMICAL  KINETICS  325 

(1)  Counter  Reactions.     In  the  chemical  change  represented  by 
the  equation 

•CHaCOOH  +  C2H5OH  <=±  CH3COOC2H5  +  H20, 

the  speed  of  the  direct  reaction  steadily  diminishes  owing  to  the 
ever-increasing  effect  of  the  reverse  or  counter  reaction.  Ulti- 
mately, when  two-thirds  of  the  acid  and  alcohol  are  decomposed, 
the  velocities  of  the  two  reactions  become  equal  and  a  condition 
of  equilibrium  results.  Starting  with  1  mol  of  acid  and  1  mol 
of  alcohol,  and  letting  x  represent  the  amount  of  ester  formed, 
we  have 

^  =  k  (1  -  x)2  -  k'x2  - 
When  equilibrium  is  attained, 


By  observing  the  change  for  any  time  t,  we  have 

7       ,,       3  ,        2  -x 
k-k'  =- 


Having  the  values  of  k/kr  and  k  —  k',  the  velocity  constant  k 
of  the  direct  reaction  can  be  determined.  The  value  of  k  so 
obtained  has  been  shown  by  Knoblauch  *  to  vary  in  those  reac- 
tions where  the  concentration  of  the  hydrogen  ion  changes. 

(2)  Side  Reactions.  When  the  same  substances  are  capable  of 
reacting  in  more  than  one  way  with  the  formation  of  different 
products,  the  several  reactions  proceed  side  by  side.  Thus, 
benzene  and  chlorine  may  react  in  two  ways  as  shown  by  the 
equations, 

(1)  C6H6  +  C12  =  C6H5C1  +  HC1, 
and 

(2)  C6H6  +  3  Cla  =  C6H6C16. 


It  is  generally  possible  to  regulate  the  conditions  under  which 
the  substances  react  so  as  to  promote  one  reaction  and  retard  the 
other. 

*  Zeit.  phys.  Chem.,  22,  268  (1897). 


326  THEORETICAL  CHEMISTRY 

(3)  Consecutive  Reactions.  By  consecutive  reactions  we  under- 
stand those  reactions  in  which  the  products  of  a  certain  initial 
chemical  change  react,  either  with  each  other  or  with  the  original 
substances  to  form  new  substances.  Attention  has  already  been 
called  to  the  fact  that  many  of  our  common  chemical  equations 
really  represent  the  summation  of  a  number  of  consecutive  reac- 
tions. If  the  system  A  is  transformed  into  the  system  C  through 
an  intermediate  system  B,  then  we  shall  have  the  two  reactions 

(1)  A^B, 
and 

(2)  £->C. 

If  reaction  (1)  should  have  a  very  much  greater  velocity  than 
reaction  (2),  then  the  measured  velocity  of  the  change  from  A  to 
C  will  be  practically  the  same  as  that  of  the  slower  reaction. 
This  fact  has  been  illustrated  by  means  of  the  following  analogy, 
due  to  James  Walker:  —  *  "The  time  occupied  by  the  transmission 
of  a  telegraphic  message  depends  both  on  the  rate  of  transmission 
along  the  conducting  wire,  and  on  the  rate  of  progress  of  the 
messenger  who  delivers  the  telegram;  but  it  is  obviously  this 
last,  slower  rate  that  is  of  really  practical  importance  in  determin- 
ing the  time  of  transmission."  The  saponification  of  ethyl 
succinate  may  be  taken  as  an  illustration  of  consecutive  reactions. 
This  reaction  proceeds  in  two  stages  as  follows:  — 


.B  /COOC2H6 

(1)  C2H4/  +  NaOH<=>C2H4/  +C2H6OH, 

XCOOC2H6  XCOONa 

/COOC2H5  xCOONa 

(2)  C2H4/  +  NaOH  <=»  C2H4  /  +  C2H5OH. 

XCOONa  XCOONa 

In  this  case  the  product  of  the  first  reaction  reacts  with  one  of  the 
original  substances. 

Velocity  of  Heterogeneous  Reactions.     It  has  been  shown  that 
when  a  solid,  such  as  calcium  carbonate,  is  dissolved  in  an  acid, 
*  Proc.  Roy.  Soc.,  Edinburgh,  22  (1898). 


CHEMICAL  KINETICS  327 

the  rate  of  solution  is  dependent  upon  the  surface  of  contact 
between  the  solid  and  liquid  phases,  and  also  upon  the  strength 
of  the  acid.  If  the  surface  is  large  so  that  it  undergoes  relatively 
little  change  during  the  reaction,  it  may  be  considered  as  constant. 
If  S  represents  the  area  of  the  surface  exposed  and  x  denotes  the 
amount  of  solid  dissolved  in  the  time  t,  the  velocity  of  the  reaction 
will  be  represented  by  the  differential  equation 

dx      7  0  ,          , 
-^  =  kS(a-  x). 

Integrating  this  equation,  we  have 


t 

This  formula  has  been  tested  by  Boguski  *  for  the  reaction 
CaC03  +  2  HC1  =  CaCl2  +  C02  +  H20, 

and  is  found  to  give  constant  values  of  k.  Furthermore,  Noyes  and 
Whitney  f  have  shown  that  the  rate  of  solution  of  a  solid  in  a  liquid 
at  any  instant,  is  proportional  to  the  difference  between  the  con- 
centration of  the  saturated  solution  and  the  concentration  of  the 
solution  at  the  time  of  the  experiment. 

Velocity  of  Reaction  and  Temperature.  It  is  a  well-estab- 
lished fact  that  the  velocity  of  a  chemical  reaction  is  accelerated 
by  rise  of  temperature.  Thus,  the  rate  of  inversion  of  cane  sugar 
is  increased  about  five  times  for  a  rise  in  temperature  of  30°.  It 
has  been  shown  as  the  result  of  a  large  number  of  observations  on 
a  variety  of  chemical  reactions,  that  in  general  the  velocity  of  a 
reaction  is  doubled  or  trebled  for  an  increase  in  temperature  of 
10°.  It  is  of  interest  to  note  that  the  rate  of  development  of 
various  organisms,  such  as  yeast  cells,  the  rate  of  growth  of  the 
eggs  of  certain  fishes,  and  the  rate  of  germination  of  certain 
varieties  of  seeds  is  either  doubled  or  trebled  for  a  rise  in  temper- 
ature of  10°.  Up  to  the  present  time  no  wholly  satisfactory  form- 
ula, connecting  the  rate  of  reaction  with  the  temperature,  has  been 
derived,  although  several  purely-empirical  expressions  have  been 

*  Berichte,  9,  1646  (1876). 

t  Zeit.  phys.  Chem.,  23,  689  (1897). 


328 


THEORETICAL  CHEMISTRY 


suggested.  Of  these  formulas  the  most  widely  applicable  is  that 
proposed  by  Van't  Hoff  and  verified  by  Arrhenius.  If  fc0  and  k\ 
represent  the  velocity  constants  at  the  respective  temperatures 
To  and  Ti,  then 

A 


where  e  is  the  base  of  the  Naperian  system  of  logarithms  and  A  is 
a  constant.  The  following  table  gives  the  calculated  and  observed 
values  of  k  at  various  temperatures  for  the  reaction 


NH4CNO<=»OC 


when  T  =  273°  +  25°,  k  =  0.000227  and  A  =  11,700. 


T, 
Degrees 

K  (observed). 

k  (calculated). 

273  +  39 

0.00141 

0.00133 

273  +  50.1 

0.00520 

0.00480 

273  +  64.5 

0.0228 

0.0227 

273  +  74.7 

0.062 

0.0623 

273  +  80 

0.100 

0.105 

In  this  case  the  agreement  between  the  observed  and  calculated 
values  is  all  that  could  be  desired. 

Influence  of  the  Solvent  on  the  Velocity  of  Reaction.  The 
velocity  of  a  chemical  reaction  varies  greatly  with  the  nature  of 
the  medium  in  which  it  takes  place.  This  subject  has  been 
studied  by  Menschutkin  *  who  has  collected  much  valuable  data, 
as  the  result  of  a  large  number  of  experiments,  on  the  velocity  of 
the  reaction  between  ethyl  iodide  and  triethylamine,  as  represented 
by  the  equation 

C2H5I  +  (C2H5)3N  =  (C2H5)4NL 

This  reaction  was  allowed  to  take  place  in  a  large  number  of 

different  solvents  and  the  velocity  at  100°  was  measured.     A  few 

*  Zeit.  phys.  Chem.,  6,  41  (1890). 


CHEMICAL  KINETICS 


329 


of  Menschutkin's  results  are  given  in  the  accompanying  table,  in 
which  k  denotes  the  velocity  constant :  — 


Medium  . 

k 

Medium. 

k 

Hexane 

0.00018 

Ethyl  alcohol  

0.0366 

Ethyl  ether 

0.000757 

Methyl  alcohol  

0.0516 

Benzene 

0.00584 

Acetone  

0.0608 

These  figures  show  that  the  velocity  of  the  reaction  is  greatly 
modified  by  the  nature  of  the  medium  in  which  it  takes  place,  the 
velocity  in  hexane  being  less  than  one  three-hundredth  of  that  in 
acetone.  It  is  of  interest  to  note  that  there  is  an  approximate 
parallelism  between  the  values  of  k,  and  the  values  of  the  dielec- 
tric constant  of  the  different  media. 

Catalysis.  It  is  a  familiar  fact  that  the  velocity  of  reaction 
is  frequently  greatly  accelerated  by  the  presence  of  a  foreign  sub- 
stance which  apparently  does  not  participate  in  the  reaction,  and 
which  remains  unchanged  when  the  reaction  is  complete.  For 
example,  cane  sugar  is  inverted  very  slowly  by  pure  water  alone, 
but  when  a  trace  of  acid  is  added  the  reaction  is  greatly  acceler- 
ated. A  substance  which  is  capable  of  exerting  such  an  acceler- 
ating action  is  termed  a  catalyst,  and  the  process  is  known  as 
catalysis.  In  addition  to  the  fact  that  a  relatively-small  amount 
of  a  catalyst  is  capable  of  effecting  the  transformation  of  large 
amounts  of  material,  there  are  two  other  important  character- 
istics of  catalytic  action  which  should  be  mentioned:  viz.,  (a)  a 
catalyst  does  not  initiate  a  reaction  but  simply  promotes  it;  and 
(6)  the  equilibrium  is  not  disturbed  by  the  presence  of  a  catalyst, 
since  the  velocities  of  the  direct  and  reverse  reactions  are  each 
altered  to  the  same  extent.  As  the  result  of  a  series  of  experi- 
ments, Ostwald  concludes  that  the  catalytic  effect  of  acids  in 
hastening  the  inversion  of  cane  sugar  is  directly  proportional  to 
the  concentration  of  the  hydrogen  ion,  and,  in  general,  is  inde- 
pendent of  the  nature  of  the  anion.  Similarly,  the  catalytic  action 
of  bases  may  be  attributed  to  the  hydroxyl  ion,  the  effect  being 
proportional  to  the  concentration  of  this  ion.  In  fact  we  may 


330  THEORETICAL  CHEMISTRY 

formulate  the  following  fundamental  law  of  catalysis  :  —  The 
degree  of  catalytic  action  is  directly  proportional  to  the  concentration 
of  the  catalytic  agent.  Almost  every  chemical  reaction  can  be 
accelerated  by  the  addition  of  an  appropriate  catalyst.  A  few 
typical  reactions  which  are  accelerated  catalytically  are  here 
given,  together  with  the  catalyst  employed  :  — 

Catalyst  —  hydrogen  ion, 

CH3COOC2H5  +  H20  =  CHaCOOH  +  C2H5OH, 

Catalyst  —  hydroxyl  ion, 

2CH3.COCH3  =  CH3.CO.CH2C(CH3)20H, 

Catalyst  —  finely  divided  platinum, 
2SO2  +  O2  =  2SO3, 
2  CH3OH  +  02  =  2  H-COH  +  2  H2O, 
2  H2  +  02  =  2  H20, 

Catalyst  —  water  vapor, 

2  CO  +  O2  =  2  CO2, 
NH4C1  =  NH3  +  HC1, 

Catalyst  —  copper  sulphate, 


(  Deacon  Process) 

Catalyst  —  mercury  salts, 

OH  +  ^  H,0  +  4  CO, 


2  C10H8  +  9  02  = 

COOH 

(First  step  in  the  synthesis  of  indigo) 

Catalyst  —  colloidal  platinum, 

2H202  =  2H20  +  02, 

Catalyst  —  enzymes, 

C6Hi206  =  2  C2H5OH  +  2  C02, 

(zymase) 

C3H7COOH  +  C2H3OH  =  C3H7COOC2H5  +  H2O. 

(lipase) 


CHEMICAL  KINETICS  331 

It  will  be  seen  that  catalysis  is  of  great  importance  in  connection 
with  many  industrial  processes  as  well  as  in  the  field  of  pure 
chemistry.  The  majority  of  the  reactions  occurring  within 
living  organisms  are  accelerated  catalytically  by  unorganized 
ferments  or  enzymes.  Thus,  before  the  process  of  digestion  can 
proceed,  starch  must  be  changed  into  sugar.  This  transformation 
is  accelerated  by  an  enzyme  called  ptyalin  occurring  in  the  saliva, 
and  by  other  enzymes  found  in  the  pancreatic  juice.  The  digestion 
of  albumen  is  hastened  by  the  enzymes,  pepsin  and  trypsin.  As 
a  rule  each  enzyme  acts  catalytically  on  just  one  reaction,  or  in 
other  words  the  catalytic  action  of  enzymes  is  specific.  Enzymes 
are  very  sensitive  to  traces  of  certain  toxic  substances  such  as 
hydrocyanic  acid,  iodine,  and  mercuric  chloride. 

An  interesting  series  of  experiments  by  Bredig  *  on  the  catalytic 
action  of  colloidal  metals,  established  the  fact  that  these  substan- 
ces resemble  the  enzymes  very  closely  in  their  behavior.  Thus, 
they  are  "poisoned"  by  the  same  substances  which  inhibit  the 
activity  of  the  enzymes,  and  they  show  the  same  tendency  to 
recover  when  the  amount  of  the  poison  does  not  exceed  a  certain 
limiting  value.  Because  of  this  close  similarity,  Bredig  called  the 
colloidal  metals  inorganic  ferments. 

It  sometimes  happens  that  one  of  the  products  of  a  chemical 
reaction  functions  as  a  catalyst  to  the  reaction.  Thus,  when 
metallic  copper  is  dissolved  in  nitric  acid,  the  reaction  proceeds 
slowly  at  first  and  then,  after  a  short  interval,  the  speed  of  the 
reaction  is  greatly  augmented.  The  acceleration  is  due  to  the 
catalytic  action  of  the  nitric  oxide  evolved.  This  phenomenon 
is  known  as  autocatalysis.  In  reactions  where  autocatalysis 
occurs,  the  velocity  increases  with  the  time  until  a  certain  maximum 
value  is  reached,  after  which  the  velocity  steadily  diminishes.  In 
ordinary  reactions  the  initial  velocity  is  the  greatest. 

It  sometimes  happens  that  the  speed  of  a  reaction  is  retarded 
by  the  presence  of  a  trace  of  some  foreign  substance.  Thus, 
Bigelow  f  has  shown  that  the  rate  of  oxidation  of  sodium  sulphite 
is  retarded  by  the  presence  in  the  solution  of  only  one  one-hundred- 

*  Zeit.  phys.  Chem.,  31,  258  (1899). 
t  Zeit.  phys.  Chem.,  26,  493  (1898). 


332  THEORETICAL  CHEMISTRY 

and-sixty-thousandth  of  a  formula  weight  of  mannite  per  liter. 
Such  a  substance  is  termed  a  negative  catalyst. 

Mechanism  of  Catalysis.  As  to  the  cause  of  catalytic  action 
very  little  is  known.  In  fact  it  is  more  reasonable  to  suppose  that 
the  mechanism  of  catalysis  varies  with  the  nature  of  the  reaction 
and  the  nature  of  the  catalyst,  than  to  conceive  all  catalytic  effects 
to  be  traceable  to  a  common  origin.  One  of  the  earliest  hypotheses 
as  to  the  mechanism  of  catalysis  was  put  forward  by  Liebig.  He 
suggested  that  the  catalyst  sets  up  intramolecular  vibrations  which 
assist  chemical  reaction.  The  vibration  theory  was  gradually 
abandoned  as  its  inadequacy  came  to  be  recognized.  Of  the  many 
explanations  which  have  been  offered  to  account  for  catalytic 
acceleration,  that  involving  the  formation  of  hypothetical  inter- 
mediate compounds  with  the  catalyst  has  been  accepted  with  the 
greatest  favor.  Thus,  if  a  reaction  represented  by  the  equation 

A  +  B  =  AB, 

takes  place  very  slowly  under  ordinary  conditions,  it  is  possible 
to  accelerate  its  velocity  by  the  addition  of  an  appropriate  cat- 
alyst C.  According  to  the  theory  of  intermediate  compounds, 
the  catalyst  is  supposed  to  act  in  the  following  manner:  — 

(1)  A  +  C  =  AC, 

(2)  AC  +  B  =  AB  +  C. 

As  will  be  seen,  the  catalyst  is  regenerated  in  the  second  stage 
of  the  reaction.  In  1806  Clement  and  Desormes  suggested  that 
the  action  of  nitric  oxide  in  promoting  the  oxidation  of  sulphur 
dioxide  in  the  manufacture  of  sulphuric  acid  was  purely  catalytic. 
As  is  well  known,  the  rate  of  the  reaction  represented  by  the 
equation 

02  =  2SO3, 


is  very  slow.     The  accelerating  action  of  nitric  oxide  on  the 
reaction  may  be  represented  in  the  following  manner:  — 


(1)  2NO  +  02  =  2NO2, 
and 

(2)  S02  +  NO2  =  S03  +  NO. 


CHEMICAL  KINETICS  333 

This  explanation,  first  offered  by  Clement  and  Desormes,  is  still 
regarded  as  the  most  plausible  explanation  of  the  part  played  by 
the  oxides  of  nitrogen  in  the  synthesis  of  sulphuric  acid.  It  is 
apparent  that  this  so-called  explanation  is  far  from  complete.  In 
fact,  it  must  be  admitted  that  we  have  no  adequate  explanation 
of  the  phenomenon  of  catalysis.  When  we  are  able  to  answer 
the  question — "Why  does  a  chemical  reaction  take  place?" — 
then  we  may  be  able  to  explain  the  accelerating  and  retarding 
influences  of  certain  foreign  substances  on  the  speed  of  reactions. 
Ostwald  likens  the  action  of  a  catalyst  to  that  of  a  lubricant  on  a 
machine  —  it  helps  to  overcome  the  resistance  of  the  reaction. 
If  the  velocity  of  a  reaction  is  represented  by  an  equation  similar 
to  that  expressing  Ohm's  law,  we  have 

,     .,       »         , .          driving  force 

velocity  of  reaction  = r-f 

resistance 

The  driving  force  is  the  same  thing  as  the  free  energy  or  chemical 
affinity  of  the  reacting  substances;  of  the  resistance  we  know 
practically  nothing.  The  velocity,  according  to  the  above  expres- 
sion, can  be  increased  in  either  of  two  ways,  viz.,  (1)  by  increas- 
ing the  driving  force,  or  (2)  by  diminishing  the  resistance.  It  is 
inconceivable  that  a  catalyst  can  exert  any  effect  upon  the  chem- 
ical affinity  of  the  reacting  substances,  so  that  we  are  forced  to 
conclude  that  its  action  must  be  confined  to  lessening  the 
resistance.* 

PROBLEMS. 

1.  When  a  solution  of  dibromsuccinic  acid  is  heated,  the  acid  decom- 
poses into  brom-maleic  acid  and  hydrobromic  acid  according  to  the 
equation 

CHBr-COOH  CH-COOH 
I         =11       +  HBr. 
CHBr-COOH   CBr-COOH 

*  For  an  excellent  review  of  the  subject  of  catalysis  the  student  is  advised 
to  consult  "Die  Lehre  von  der  Reaktionsbeschleunigung  durch  Fremdstoffe," 
by  W.  Herz.  Ahrens'  "Sammlung  chemischer  and  chemisch-technischer 
Vortraege." 


334 


THEORETICAL  CHEMISTRY 


At  50°  the  initial  titre  of  a  definite  volume  of  the  solution  was  T0  = 
10.095  cc.  of  standard  alkali.  After  t  minutes  the  titre  of  the  same 
volume  of  solution  was  Tt  cc.  of  standard  alkali. 

t  0  214  380 

Tt  10.095  10.37  10.57 

(a)  Calculate  the  velocity-constant  of  the  reaction. 

(b)  After  what  time  is  one-third  of  the  dibromsuccinic  acid  decom- 
posed? Am.    (a)  0.000260;  (b)  1559  minutes. 

2.  From  the  following  data  show  that  the  decomposition  of  H202  in 
aqueous  solution  is  a  unimolecular  reaction:  — 

Time  in  minutes  0  10  20 

n  22. 8          13.8  8.25cc.' 

n  is  the  number  of  cubic  centimeters  of  potassium  permanganate  required 
to  decompose  a  definite  volume  of  the  hydrogen  peroxide  solution. 

3.  In  the  saponification  of  ethyl  acetate  by  sodium  hydroxide  at  10°, 
y  cc.  of  0.043  molar  hydrochloric  acid  were  required  to  neutralize  100  cc. 
of  the  reaction  mixture  t  minutes  after  the  commencement  of  the  reaction. 

t  0  4.89          10.37          28.18        infinity 

y  61.95          50.59          42.40          29.35        14.92 

Calculate  the  velocity-constant  when  the  concentrations  are  expressed 
in  mols  per  liter.  Ans.  Mean  value  of  k  =  2.38. 

4.  The  velocity-constant  of  formation  of  hydriodic  acid  from  its  ele- 
ments is  0.00023;   the  equilibrium  constant  at  the  same  temperature  is 
0.0157.     What  is  the  velocity-constant  of  the  reverse  reaction? 

Ans.  0.0146. 

5.  Determine  the  order  of  the  following  reaction :  — 

6  FeCl2  +  KC103  +  6  HC1  =  6  FeCl3  +  KC1  +  3  H20. 
When  the  initial  concentration  of  the  reacting  substances  is  0.1,  the 
changes  in  concentration  at  successive  times  are  as  follows :  — 


Time  (minutes). 

Change  in  Con- 
centration. 

5 

0.0048 

15 

0.0122 

35 

0.0238 

60 

0.0329 

110 

0.0452 

170 

0.0525 

Ans.  Third  order. 


CHAPTER   XV. 
ELECTRICAL   CONDUCTANCE. 

Historical  Introduction.  In  a  book  of  this  character  it  is 
impossible  to  give  anything  like  a  complete  historical  sketch  of 
electrochemistry.  Before  entering  upon  an  outline  of  this  inter- 
esting division  of  theoretical  chemistry,  however,  it  is  desirable 
to  consider  very  briefly  a  few  of  the  theories  which  have  played  a 
prominent  part  in  the  development  of  our  modern  views  concern- 
ing electrochemical  phenomena.  While  the  early  observations  of 
Beccaria  and  others  pointed  to  the  probability  of  the  existence 
of  some  relation  between  chemical  and  electrical  phenomena,  it 
was  not  until  the  beginning  of  the  nineteenth  century  that  the 
science  of  electrochemistry  had  its  birth.  The  epoch-making 
discovery  by  Volta  of  a  means  of  obtaining  electrical  energy  from 
chemical  energy,  gave  the  initial  impulse  to  all  the  brilliant  dis- 
coveries and  investigations  upon  which  the  modern  science  of 
electrochemistry  is  based.  The  apparatus  devised  by  Volta, 
known  as  the  voltaic  pile,  consisted  of  disks  of  zinc  and  silver 
placed  alternately  over  one  another,  the  silver  disk  of  one  pair 
being  separated  from  the  zinc  disk  of  the  next  by  a  piece  of 
blotting  paper  moistened  with  brine.  Such  a  pile,  if  composed 
of  a  sufficient  number  of  pairs  of  disks,  will  produce  electricity 
enough  to  give  a  shock,  if  the  top  and  bottom  disks,  or  wires 
connected  with  them,  be  touched  with  the  moist  fingers.  This 
discovery  placed  in  the  hands  of  the  investigator  a  source  of 
electricity  by  means  of  which  experiments  could  be  performed 
which  had  hitherto  been  impossible.  Shortly  after  the  discovery 
of  the  voltaic  pile,  Nicholson  and  Carlisle  *  effected  the  decom- 
position of  water,  and  Davy  f  isolated  the  alkali  metals.  As 
a  result  of  these  experiments,  Davy  was  led  to  formulate  his 

*  Nich.  Jour.,  4,  179  (1800). 

t  Ibid.,  4,  275,  326  (1800);  Gilb.  Ann.,  7,  114  (1801). 
335 


336  THEORETICAL  CHEMISTRY 

electrochemical  theory.  According  to  this  theory,  the  atoms  of 
different  substances  acquire  opposite  electrical  charges  by  con- 
tact, and  thus  mutually  attract  each  other.  If  the  differences 
between  the  charges  are  small,  the  attraction  will  be  insufficient 
to  cause  the  atoms  to  leave  their  former  positions;  if  it  is  great, 
a  rearrangement  of  the  atoms  will  occur  and  a  chemical  com- 
pound will  be  formed.  In  terms  of  this  theory,  electrolysis  con- 
sists in  a  neutralization  of  the  charges  upon  the  atoms. 

The  theory  of  Davy  was  soon  superseded  by  that  of  Berzelius.* 
According  to  the  latter  theory,  every  atom  is  charged  with  both 
kinds  of  electricity  which  exist  upon  the  atoms  in  a  polar  arrange- 
ment, the  electrical  behavior  of  the  atom  being  determined  by  the 
kind  of  electricity  which  is  in  excess.  Chemical  attraction  is  merely 
the  electrical  attraction  of  oppositely-charged  atoms.  Since  each 
atom  is  endowed  with  both  positive  and  negative  electrification, 
one  charge  being  in  excess,  it  follows  that  the  compound  formed 
by  the  union  of  two  or  more  atoms  will  be  positively  or  negatively 
charged  according  to  whichever  charge  remains  unneutralized  after 
the  atoms  have  combined.  Two  compounds,  the  one  charged  pos- 
itively and  the  other  negatively,  may  thus  in  turn  combine,  a 
more  complex  compound  being  formed.  Shortly  after  Berzelius 
formulated  his  theory,  it  became  the  subject  of  much  discussion 
and  was  severely  criticized.  Thus,  it  was  pointed  out  that  if 
chemical  combination  results  from  the  neutralization  of  oppo- 
sitely-charged atoms,  then  as  soon  as  the  charges  have  become 
equalized,  there  no  longer  exists  any  attractive  force  and  the  com- 
pound must  again  decompose.  This  objection  was  easily  overcome 
by  assuming  that  as  soon  as  the  union  between  the  atoms  is 
broken,  they  again  acquire  their  original  charges  and,  in  conse- 
quence, recombine.  In  other  words,  a  chemical  compound  is  to 
be  regarded  as  existing  in  a  state  of  unstable  equilibrium.  An- 
other, and  apparently  insurmountable,  objection  to  the  theory 
resulted  from  the  exceptions  presented  by  acetic  acid  and  some  of 
its  substitution  products. 

According  to  the  theory  of  Berzelius,  chemical  combination  is 
entirely  dependent  upon  the  nature  of  the  electrical  charges  resid- 
*  Gilb.  Ann.,  27,  270  (1807). 


ELECTRICAL   CONDUCTANCE  337 

ing  on  the  atoms.  From  this  statement  it  follows  that  the  prop- 
erties of  a  chemical  compound  must  be  a  function  of  the  electrical 
charges  upon  the  atoms  of  its  constituents.  It  was  shown  that 
when  the  three  hydrogen  atoms  of  the  methyl  group  in  acetic 
acid  are  successively  replaced  by  chlorine,  the  chemical  properties 
of  the  original  substance  are  not  materially  altered.  According 
to  Berzelius,  the  three  hydrogen  atoms  are  positively  charged 
while  the  three  chlorine  atoms  are  negatively  charged.  That 
three  negative  charges  could  be  substituted  for  three  positive 
charges  in  acetic  acid  without  producing  a  more  marked  change 
in  its  properties,  could  not  be  satisfactorily  accounted  for  by  the 
theory.  This  criticism  was  for  a  long  time  considered  as  an 
insuperable  barrier  to  the  acceptance  of  the  theory.  Shortly 
before  the  close  of  the  nineteenth  century,  J.  J.  Thomson  *  showed 
that  this  objection  has  little  or  no  weight.  When  hydrogen  gas 
is  electrolyzed  in  a  vacuum-tube  and  the  spectra  at  the  two  elec- 
trodes are  compared,  Thomson  found  them  to  differ  widely. 
From  this  he  concluded  that  the  molecule  of  hydrogen  gas  is  in 
all  probability  made  up  of  positively-  and  negatively-charged  parts 
or  ions.  He  then  extended  his  experiments  to  the  vapors  of  cer- 
tain organic  compounds.  In  discussing  these  experiments  he 
says:  —  "In  many  organic  compounds,  atoms  of  an  electro- 
positive element,  hydrogen,  are  replaced  by  atoms  of  an  elec- 
tronegative element,  chlorine,  without  altering  the  type  of  the 
compound.  Thus,  for  example,  we  can  replace  the  four  hydrogen 
atoms  in  CH4  by  chlorine  atoms,  getting,  successively,  the  com- 
pounds CH3C1,  CH2C12,  CHCla,  and  CC14.  It  seemed  of  interest 
to  investigate  what  was  the  nature  of  the  charge  of  electricity  on 
the  chlorine  atoms  in  these  compounds.  The  point  is  of  some 
historical  interest,  as  the  possibility  of  substituting  an  electro- 
negative element  in  a  compound  for  an  electropositive  one  was 
one  of  the  chief  objections  against  the  electrochemical  theory  of 
Berzelius." 

"  When  the  vapor  of  chloroform  was  placed  in  the  tube,  it  was 
found  that  both  the  hydrogen  and  chlorine  lines  were  bright  on 
the  negative  side  of  the  plate,  while  they  were  absent  from  the 
*  Nature,  52,  451  (1895). 


338  THEORETICAL  CHEMISTRY 

positive  side,  and  that  any  increase  in  brightness  of  the  hydrogen 
lines  was  accompanied  by  an  increase  in  the  brightness  of  those 
due  to  chlorine.  The  appearance  of  the  hydrogen  and  chlorine 
spectra  on  the  same  side  of  the  plate  was  also  observed  in  methy- 
lene  chloride  and  in  ethylene  chloride.  Even  when  all  the 
hydrogen  in  methane  was  replaced  by  chlorine,  as  in  carbon  tetra- 
chloride,  the  chlorine  spectra  still  clung  to  the  negative  side  of 
the  plate.  The  same  point  was  tested  with  silicon  tetrachloride 
and  the  chlorine  spectrum  was  brightest  on  the  negative  side  of 
the  plate.  From  these  experiments  it  would  appear,  that  the 
chlorine  atoms  in  the  chlorine  derivatives  of  methane  are  charged 
with  electricity  of  the  same  sign  as  the  hydrogen  atoms  they 
displace." 

Electrical  Units.  In  1827,  Dr.  G.  S.  Ohm  enunciated  his  well- 
known  law  of  electrical  conductance,  viz. :  —  The  strength  of  the 
electric  current  flowing  in  a  conductor  is  directly  proportional  to  the 
difference  of  potential  between  the  ends  of  the  conductor,  and  inversely 
proportional  to  its  resistance.  If  C  represents  the  strength  of  the 
current,  E  the  difference  of  potential,  and  R  the  resistance,  then 
Ohm's  law  may  be  formulated  thus:  — 

E 

r  —  —. 

-R 

The  unit  of  resistance  is  the  ohm,  that  of  difference  of  potential 
or  electromotive  force,  the  volt,  and  that  of  current,  the  ampere. 
The  ohm  is  denned  as  the  resistance  of  a  column  of  mercury 
106.3  cm.  long  and  1  sq.  mm.  in  cross  section  at  0°  C.  The 
ampere  is  denned  as  the  current  which  will  cause  the  deposition 
of  0.001118  gram  of  silver  from  a  solution  of  silver  nitrate  in  1 
second.  The  volt  may  be  defined  as  the  electromotive  force 
necessary  to  drive  a  current  of  1  ampere  through  a  resistance  of 
1  ohm.  The  unit  of  quantity  of  electricity  is  the  coulomb,  and  is 
equivalent  to  a  current  of  1  ampere  per  second.  One  gram 
equivalent  of  any  ion  carries  96,540  coulombs,  a  quantity  of 
electricity  known  as  the  faraday  =  F.  As  has  already  been 
pointed  out,  any  form  of  energy  may  be  considered  as  the  product 
of  two  factors,  a  capacity  factor  and  an  intensity  factor. 


ELECTRICAL   CONDUCTANCE  339 

The  capacity  factor  of  electrical  energy  is  the  coulomb  while 
the  intensity  factor  is  the  volt,  i.e., 

electrical  energy  =  coulombs  X  volts. 

The  unit  of  electrical  energy,  therefore,  is  the  volt-ampere-second 
commonly  called  the  watt-second.  One  watt-second  is  the  elec- 
trical work  done  by  a  current  of  1  ampere  flowing  under  an  elec- 
tromotive force  of  1  volt  for  1  second,  and  is  equivalent  to  1  X  107 
C.G.S.  units.  The  thermal  equivalent  of  electrical  energy  may  be 
calculated  from  the  relation 

electrical  energy  in  absolute  units       ,  £    , 

.   °J : 7-r — T =  heat  equiv.  of  elect,  energy, 

mechanical  equiv.  of  heat 

or 

1X107 


42,600  X  980.1 


0.2394  cal.  =  1  watt-second. 


Faraday's  Laws.  When  two  platinum  plates  or  electrodes,  one 
connected  to  the  positive  and  the  other  to  the  negative  terminal 
of  a  battery,  are  immersed  in  a  solution  of  sodium  chloride,  it 
will  be  found  that  hydrogen  is  immediately  evolved  at  the  nega- 
tive electrode  and  oxygen  at  the  positive  electrode.  If  the  salt 
solution  is  previously  colored  with  a  few  drops  of  a  solution  of 
litmus  it  will  be  observed  that  the  portion  of  the  solution  in  the 
neighborhood  of  the  positive  electrode  will  turn  red,  indicating 
the  formation  of  an  acid,  while  that  in  the  neighborhood  of  the 
negative  electrode  will  turn  blue,  showing  the  formation  of  a 
base.  The  same  changes  will  take  place  whether  the  electrodes 
are  placed  near  together  or  far  apart,  and  furthermore,  the  evolu- 
tion of  gas  and  the  change  in  color  at  the  electrodes  commences 
as  soon  as  the  circuit  is  closed.  The  study  of  these  'phenomena 
led  Faraday  *  to  the  conclusion,  that  when  an  electric  current 
traverses  a  solution,  there  occurs  an  actual  transfer  of  matter, 
one  portion  travelling  with  the  current  and  the  other  portion 
moving  in  the  opposite  direction.  At  the  suggestion  of  the  philol- 
ogist Whewell,  Faraday  termed  these  carriers  of  the  current,  ions 
=  to  wander).  He  also  called  the  electrode  connected  to 
*  Experimental  Researches,  (1834). 


340 


THEORETICAL  CHEMISTRY 


the  positive  terminal  of  the  battery,  the  anode,  (ava  =  up  and 
68os  =  way),  and  the  electrode  connected  to  the  negative  terminal 
the  cathode,  (Kara.  —  down  and  68os  =  way).  The  ions  which 
move  toward  the  anode  he  called  anions,  while  those  which  migrate 
toward  the  cathode  he  called  cations.  The  whole  process  he- 
termed  electrolysis.  The  question  of  the  relationship  between  the 
amount  of  electrolysis  and  the  quantity  of  electricity  passing 
through  a  solution  was  investigated  by  Faraday.  As  a  result  of 
his  experiments  he  enunciated  the  following  laws  which  are  com- 
monly known  as  the  laws  of  Faraday :  — 

(1)  For  the  same  electrolyte,  the  amount  of  electrolysis  is  propor- 
tional to  the  quantity  of  electricity  which  passes. 

(2)  The  amounts  of  substances  liberated  at  the  electrodes  when  the 
same  quantity  of  electricity  passes  through  solutions  of  different 
electrolytes,   are  proportional   to   their  chemical  equivalents.     The 
chemical  equivalent  of  any  ion  is  equal  to  the  atomic  weight  divided 
by  its  valence.     If  the  same  quantity  of  electricity  is  passed 
through  solutions  of  hydrochloric  acid,  silver  nitrate,  cuprous 
chloride,  cupric  chloride,  and  auric  chloride,  the  relative  amounts 
of  the  different  cations  liberated  will  be  as  follows:  — 


Electrolyte. 

Chem.  Equiv.  of 
Cation. 

HC1. 

AgN03 
Cu2Cl2 
CuCl2 
AuCl3 

H'  =  l 

Ag'  =  10S 
Cu'  =  63.4 
Cu"  =  63.4-r-2 
Au—  =  197+3 

The  electrochemical  equivalent  of  an  element  or  group  of  elements 
is  the  weight  in  grams  which  is  liberated  by  the  passage  of  one 
coulomb  of  electricity.  The  electrochemical  equivalents  are, 
according  to  Faraday's  second  law,  proportional  to  the  chemical 
equivalents.  The  quantity  of  electricity  necessary  to  liberate  one 
chemical  equivalent  in  grams  is  called  a  faraday.  This  is  a  very 
important  unit  in  electrochemical  calculations.  Since  one  coulomb 
liberates  0.00001036  gram  of  hydrogen,  1  -f-  0.00001036  =  96,540 


ELECTRICAL   CONDUCTANCE  341 

coulombs  of  electricity  will  be  required  to  liberate  one  gram  equiv- 
alent of  hydrogen.  The  same  quantity  of  electricity  will  liberate 
35.45  X  0.00001036  =  0.000368  gram  of  chlorine,  and  108  X 
0.00001036  =  0.001118  gram  of  silver.  Or,  in  general,  since  one 
coulomb  of  electricity  liberates  0.00001036  gram  of  hydrogen,  it 
will  cause  the  liberation  of  0.00001036  w  grams  of  any  other  ele- 
ment whose  equivalent  weight  is  w. 

The  Existence  of  Free  Ions.  When  an  electrolyte  is  de- 
composed by  the  electric  current,  the  products  of  decomposition 
appear  at  the  electrodes.  The  fact  that  the  liberation  of  the  prod- 
ucts of  decomposition  is  independent  of  the  distance  between 
the  electrodes  caused  considerable  difficulty  in  the  early  history 
of  electrolysis.  It  was  evident  that  the  two  products  could 
hardly  be  derived  from  the  same  molecule,  but  must  come  from 
two  different  molecules.  Several  theories  were  advanced  to 
account  for  the  experimental  results.  Thus,  in  the  electrolysis  of 
water  it  was  suggested  that  the  two  gases,  hydrogen  and  oxygen, 
were  not  derived  from  the  water  but  that  electricity  itself  pos- 
sessed an  acid  character.  Grotthuss  *  was  the  first  to  propose  a 
rational  hypothesis  as  to  the  mechanism  of  electrolysis.  He 
assumed  that  when  the  electrodes  in  an  electrolytic  cell  are  con- 
nected with  a  source  of  electricity,  the  molecules  of  the  electrolyte 
arrange  themselves  in  straight  lines  between  the  electrodes,  th,e 
positive  poles  being  directed  toward  the  negative  electrode  and 
the  negative  poles  toward  the  positive  electrode.  When  elec- 
trolysis begins,  the  cation  of  the  molecule  nearest  the  cathode 
is  liberated  at  the  cathode  and  the  anion  of  the  molecule  nearest 
the  anode  is  liberated  at  the  anode.  The  anion  which  is  left 
free  near  the  cathode  then  combines  with  the  cation  of  the  next 
adjoining  molecule,  the  anion  thus  left  uncombined  uniting  with 
the  cation  of  its  nearest  neighbor,  a  similar  exchange  of  partners 
continuing  throughout  the  entire  molecular  chain.  Under  the 
directive  influence  of  the  two  electrodes,  the  newly-grouped  mole- 
cules then  rotate  so  that  the  positive  poles  all  face  the  negative 
electrode  and  the  negative  poles  all  face  the  positive  electrode. 
The  process  is  then  repeated,  another  molecule  being  electrolyzed. 
*  Ann.  de  Chim.  [1],  58,  54  (1806). 


342  THEORETICAL  CHEMISTRY 

This  theory  of  electrolysis  appears  to  have  been  accepted  by 
Faraday.  Its  inherent  defect  was  first  pointed  out  by  Grove.* 
From  his  experiments  with  the  oxy-hydrogen  cell,  which  derives 
its  energy  from  the  union  of  hydrogen  and  oxygen,  he  pointed  out 
that  a  decomposition  of  the  molecules  of  water  is  not  essential 
for  the  evolution  of  these  two  gases,  but  that  the  molecules  must 
be  already  in  a  state  of  partial  decomposition.  This  suggestion 
was  followed  up  by  Clausius.  f  He  argued  that  if  an  expenditure 
of  energy  is  necessary  to  decompose  the  molecules,  electrolysis 
should  be  impossible  at  very  low  voltages.  Experiment  showed 
that  when  silver  nitrate  is  electrolyzed  between  silver  electrodes, 
decomposition  takes  place  at  voltages  which  are  much  below  the 
voltage  corresponding  to  the  energy  of  formation  of  silver  nitrate. 
In  other  words,  it  requires  very  little  energy  to  decompose  a  salt 
which  is  formed  with  the  evolution  of  a  large  amount  of  energy, 
a  result  which  is  in  contradiction  to  the  principle  of  the  conserva- 
tion of  energy.  Clausius  was  thus  forced  to  conclude  "  that  the 
supposition  that  the  constituents  of  the  molecule  of  an  electrolyte 
are  firmly  united  and  exist  in  a  fixed  and  orderly  arrangement  is 
wholly  erroneous." 

As  a  result  of  his  investigation  of  the  synthesis  of  ethyl  ether 
from  alcohol  and  sulphuric  acid,  Williamson  J  concluded  "that  in 
an  aggregate  of  the  molecules  of  every  compound,  a  constant  inter- 
change between  the  elements  contained  in  them  is  taking  place. " 
In  the  same  paper  he  writes,  "each  atom  of  hydrogen  does  not 
remain  quietly  attached  all  the  time  to  the  same  atom  of  chlorine, 
but  they  are  continually  exchanging  places  with  one  another." 
This  view  was  accepted  by  Clausius,  although  he  had  no  means  of 
determining  the  extent  to  which  the  electrolyte  was  broken  down 
or  dissociated  into  free  ions. 

In  1887,  Arrhenius  §  developed  the  views  of  Clausius  by  showing 
how  the  degree  of  dissociation  of  the  molecules  of  an  electrolyte 
can  be  deduced  from  measurements  of  the  electrical  conductance 

*  Phil.  Mag.,  27,  348  (1845). 
t  Pogg.  Ann.,  101,  338  (1857). 
j  Lieb.  Ann.,  77,  37  (1851). 
§  Zeit.  phys.  Chem.,  i,  631  (1887). 


ELECTRICAL   CONDUCTANCE  343 

of  its  solutions,  as  well  as  from  measurements  of  osmotic  pressure 
and  freezing-point  lowering.  The  important  generalization  sum- 
marizing these  conceptions  is  known  as  the  theory  of  electrolytic 
dissociation,  to  which  reference  has  already  been  made  in  earlier 
chapters  (see  page  205). 

The  Migration  of  the  Ions.  Since  the  passage  of  a  current  of 
electricity  through  a  solution  of  an  electrolyte  causes  the  dis- 
charge of  equivalent  amounts  of  positive  and  negative  ions  at  the 
electrodes,  it  might  be  inferred  that  the  ions  all  move  with  the 
same  speed.  That  this  inference  is  incorrect,  was  first  shown  by 
Hittorf  *  as  the  result  of  his  observations  on  the  changes  in  con- 
centration in  the  neighborhood  of  the  electrodes  during  electroly- 


I 

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000  0i0  0000  OiO  000 

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O  ©  O  O  O  ©|O  O  0  O  ©  OiO  O 

in 

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O  O  O  O  O  ©JO  O  0  O  0  OiO  O 

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IV 

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O  O  O  O  O  OiO  O  ©  O  ©  OiO  O 


I 


Fig.  82. 

sis.     The  effect  of  unequal  ionic  velocities  on  the  concentrations  of 
the  solutions  around  the  electrodes  is  clearly  shown  by  the  accom- 
panying diagram   (Fig.   82)   due  to  Ostwald.     The  anode  and 
*  Pogg.  Ann.,  89,  177;  98,  1;  103,  1;  106,  337,  513  (1853-1859). 


344  THEORETICAL  CHEMISTRY 

cathode  in  an  electrolytic  cell  are  represented  by  the  vertical  lines 
A  and  C  respectively.  The  cell  is  divided  into  three  compart- 
ments by  means  of  porous  diaphragms,  represented  by  the  ver- 
tical dotted  lines.  The  cations  are  represented  by  dots  (")  and 
the  anions  by  dashes  (').  Before  the  current  passes  through  the 
cell,  the  concentration  of  the  solution  is  uniform  throughout,  the 
conditions  being  represented  by  I.  Now  let  us  imagine  that  only 
the  anions  move  when  the  current  is  established.  The  conditions 
when  the  chain  of  anions  has  moved  two  steps  toward  the  anode 
are  shown  in  II.  Each  ion  which  has  been  deprived  of  a  partner 
is  supposed  to  be  discharged.  It  will  be  observed  that  although 
the  cations  have  not  migrated  toward  the  cathode,  yet  an  equal 
number  of  positive  and  negative  ions  are  discharged,  and  that 
while  the  concentration  in  the  anode  compartment  has  not  changed, 
the  concentration  in  the  cathode  compartment  has  diminished  to 
one-half  its  original  value. 

Let  us  now  suppose  that  both  anions  and  cations  move  with  the 
same  speed,  and  as  before,  let  each  chain  of  ions  move  two  steps 
toward  their  respective  electrodes,  as  indicated  in  III.  It  will  be 
seen  that  four  positive  and  four  negative  ions  have  been  dis- 
charged, and  that  the  concentration  of  the  electrolyte  in  the  anode 
and  cathode  compartments  has  diminished  to  the  same  extent. 
Finally,  let  us  assume  that  the  ratio  of  the  speeds  of  the  cations 
to  that  of  the  anions  is  as  3  :  2.  When  the  cations  have  moved 
three  steps  toward  the  cathode  and  the  anions  have  moved  two 
steps  toward  the  anode,  the  conditions  will  be  as  shown  in  IV. 
It  is  evident  that  five  positive  and  five  negative  ions  have  been 
discharged,  and  that  the  concentration  in  the  cathode  compart- 
ment has  diminished  by  two  molecules  while  the  concentration 
in  the  anode  compartment  has  diminished  by  three  molecules. 
It  will  be  observed  that  the  change  in  concentration  in  either  of 
the  electrode  compartments  is  proportional  to  the  speed  of  the 
ion  leaving  it.  Thus,  in  II,  the  concentration  in  the  cathode 
compartment  diminishes  while  that  in  the  anode  compartment 
remains  unchanged,  since  only  the  anion  moves.  In  like  manner, 
the  change  in  concentration  about  the  electrodes  in  III  corre- 
sponds with  the  fact  that  both  ions  migrate  at  the  same  rate. 


ELECTRICAL   CONDUCTANCE  345 

In  IV  the  ratio  of  the  change  in  concentration  in  the  cathode 
compartment  to  that  in  the  anode  compartment  is  as  2  :  3.  It 
will  be  apparent  from  these  examples,  that  the  relation  between 
the  speeds  of  the  ions  and  the  corresponding  changes  in  concen- 
tration at  the  electrodes  may  be  expressed  by  the  following  pro- 
portion :  — 

Change  in  concentration  at  anode    _  speed  of  cation 
Change  in  concentration  at  cathode       speed  of  anion 

If  the  relative  speed  of  the  cations  is  represented  by  u,  and  that 
of  the  anions  by  v,  then  the  total  quantity  of  electricity  trans- 
ported will  be  proportional  to  u  +  v:  of  this  total,  the  fractions 

carried  by  the  anion  and  cation  respectively,  will  be  n  =  — -r— , 

and     1  —  n  =  — ; —     The  values  of  these  ratios,  n  and  1  —  n. 
u  +  v 

are  called  the  transport  numbers  of  the  anion  and  cation  respec- 
tively. It  is  apparent  from  the  diagram,  that  if  the  electrolysis 
is  not  carried  too  far,  the  concentration  of  the  solution  in  the  inter- 
mediate compartment  will  undergo  no  change.  In  order  to  deter- 
mine transport  numbers,  therefore,  it  is  simply  necessary  to 
remove  portions  of  the  solutions  in  the  immediate  vicinity  of  the 
two  electrodes  and  determine  the  concentration  of  the  electrolyte 
analytically.  The  success  of  the  experiment  depends  upon  keep- 
ing the  concentration  of  the  intermediate  compartment  unaltered. 
Experimental  Determination  of  Transport  Numbers.  Various 
forms  of  apparatus  have  been  constructed  for  the  determination 
of  transport  numbers,  among  which  one  of  the  most  satisfactory 
is  that  devised  by  Jones  and  Bassett,*  and  shown  in  Fig.  83.  It 
consists  of  two  vertical  tubes  of  wide  bore  connected  by  a  U-tube 
fitted  with  a  stop-cock.  Into  each  of  two  electrodes,  made  of 
a  suitable  metal,  is  riveted  a  short  piece  of  stout  platinum  wire, 
which  is  then  sealed  into  heavy-walled  glass  tubes.  The  exposed 
end  of  the  platinum  wire  on  the  under  side  of  each  electrode  is 
covered  with  a  drop  of  fusion  glass.  The  tubes  carrying  the 
electrodes  are  fitted  into  holes  bored  through  the  ground  glass 
*  Am.  Chem.  Jour.,  32,  409  (1904). 


346 


THEORETICAL  CHEMISTRY 


stoppers  which  close  the  right  and  left  arms  of  the  apparatus. 
Two  small  graduated  tubes  are  sealed  to  the  two  vertical  tubes 
just  below  the  stoppers.  These  tubes  allow  for  any  slight  dis- 
placement of  the  solution  due  to  expansion  or  the  formation  of 


Fig.  83. 

gas,  and  at  the  same  time  make  it  possible  to  level  the  apparatus 
accurately.  When  electrolysis  has  proceeded  far  enough,  the 
circuit  is  broken  and  the  stop-cock  closed,  thus  preventing  the 
mixing  of  the  solutions  in  the  anode  and  cathode  compartments. 
The  solutions  in  the  two  halves  of  the  apparatus  are  then  rinsed  out 
into  separate  beakers  and  the  concentration  of  each  is  determined 
analytically.  Knowing  the  initial  concentration  of  the  solution 
and  the  final  concentrations  at  the  two  electrodes,  together  with 
the  total  quantity  of  electricity  which  has  passed  through  the 
apparatus  during  the  experiment,  we  have  all  of  the  data  neces- 
sary for  the  calculation  of  the  transport  numbers  of  the  two  ions. 


ELECTRICAL   CONDUCTANCE  347 

The  following  example  will  serve  to  make  the  method  of  calcu- 
lation clear:  —  In  an  experiment  to  determine  the  transport 
numbers  of  the  ions  of  silver  nitrate,  a  solution  containing  0.00739 
gram  of  that  salt  per  gram  of  water  was  prepared.  The  solution 
was  introduced  into  the  migration  apparatus  and,  after  inserting 
silver  electrodes,  a  small  current  was  passed  through  the  appa- 
ratus for  two  hours.  A  silver  coulometer  was  included  in  the  cir- 
cuit, and  0.0780  gram  of  silver  was  deposited  by  the  current. 

This  mass  of  silver  is  equivalent  to  0.000723  gram-equivalent. 
After  the  circuit  was  broken,  the  anode  solution  was  rinsed  out  and 
its  concentration  determined  analytically.  It  was  found  to  con- 
tain 0.2361  gram  of  silver  nitrate  to  23.14  grams  of  water.  This 
amount  of  solution  contained  originally  23.14  X  0.00739  = 
0.1710  gram  of  silver  nitrate.  Thus,  the  amount  of  silver  nitrate 
in  the  anode  compartment  had  increased  by  0.2361  —  0.1710  = 
0.0651  gram  of  silver  nitrate,  or  0.000383  gram-equivalent  of 
silver.  Obviously  the  increase  in  the  concentration  of  the  nitrate 
ion  must  have  been  the  same.  The  amount  of  silver  dissolved 
from  the  anode  must  have  been  equal  to  that  deposited  in  the 
coulometer,  or  since  0.000723  gram-equivalent  of  silver  was 
deposited  and  the  actual  increase  found  was  0.000383  gram- 
equivalent,  the  difference,  0.000723  -  0.000383  =  0.000340  gram- 
equivalent,  is  the  amount  of  silver  which  migrated  away  from  the 
anode.  At  the  same  time  0.000383  gram-equivalent  of  nitrate 
ions  migrated  into  the  anode  compartment.  The  ratio  of  the 
speed  of  migration  of  the  silver  ions  to  that  of  the  nitrate  ions  is 
as  0.000340  :  0.000383.  Since  0.000723  gram-equivalent  of  silver 
ions  measures  the  total  quantity  of  electricity  transported,  the 
transport  numbers  of  the  two  ions  will  be  as  follows  :  — 

0.000340 
Transport  number  of  Ag  =  n  =     000723  =  0.470, 


Transport  number  of  NO/  =  1  -  n  =     ZZTir  =  0.530- 


These  numbers  can  be  checked  by  a  similar  calculation  based  on 
the  change  in  concentration  in  the  cathode  compartment. 


348 


THEORETICAL  CHEMISTRY 


The  following  table  gives  the  transport  numbers  of  the  anions 
of  various  electrolytes  at  different  dilutions,  V  being  the  number 
of  liters  of  solution  containing  one  gram-equivalent  of  solute. 
The  transport  numbers  of  the  corresponding  cations  can  be  found 
by  subtracting  the  transport  numbers  of  the  anions  from  unity. 

TRANSPORT  NUMBERS  OF  ANIONS. 


v= 

100 

50 

20 

10 

5 

2 

1 

0.5 

KC1       l 
KBr        1 

OK.f\a 

Ocny 

Deny 

Ocnc 

OKHQ 

f)     K10 

OC-M 

OKI  e 

KI 
NH4C1    J 

NaCl 

0.614 

0.617 

0.620 

0.626 

0.637 

KNO3 

0.497 

0.496 

0.492 

0.487 

0  479 

AgNO3 

0.528 

0.528 

0.528 

0.528 

0.527 

0.519 

0.501 

0.476 

KOH 

0.735 

0.736 

HC1         

0.172 

0.172 

0.172 

0.173 

0.176 

i  BaCl2  

0.640 

0.657 

f  K2CO3 

0  435 

0  434 

0  413 

£  CuSO4 

0  620 

0  626 

0  632 

0  643 

0.668 

0  696 

0  720 

\  H2SO4 

0.182 

0  174 

It  is  apparent  from  the  table  that  the  transport  numbers  are 
not  entirely  independent  of  the  concentration.  They  also  vary 
slightly  with  the  temperature  and  approach  the  limiting  value, 
0.5,  at  high  temperatures. 

Specific,  Molar  and  Equivalent  Conductance.  As  is  well 
known,  the  resistance  of  a  metallic  conductor  is  directly  propor- 
tional to  its  length  and  inversely  proportional  to  its  area  of  cross- 
section.  Similarly,  the  resistance  of  an  electrolyte  is  proportional 
to  the  length  and  inversely  proportional  to  the  cross-section  of 
the  column  of  solution  between  the  two  electrodes.  The  specific 
resistance  of  an  electrolyte  may  be  defined  as  the  resistance  in 
ohms  of  a  column  of  solution  one  centimeter  long  and  one  square 
centimeter  in  cross-section.  Specific  conductance  is  the  reciprocal 
of  specific  resistance.  Since  the  conductance  of  a  solution  is 
almost  wholly  dependent  upon  the  amount  of  solute  present,  it 
is  more  convenient  to  express  conductance  in  terms  of  the  molar 
or  equivalent  concentration.  The  molar  conductance  ft,  is  the 


ELECTRICAL   CONDUCTANCE 


349 


conductance  in  reciprocal  ohms,  of  a  solution  containing  one  mol 
of  solute  when  placed  between  electrodes  which  are  exactly  one 
centimeter  apart.  The  equivalent  conductance  A  is  the  conduc- 
tance in  reciprocal  ohms  of  a  solution  j  containing  one  gram- 
equivalent  of  solute  when  placed  between  electrodes  which  are 
one  centimeter  apart.  If  K  denotes  the  specific  conductance  of  a 
solution  and  Vmj  the  volume  in  cubic  centimeters  which  contains 
one  mol  of  solute,  then 


and  in  like  manner 


KVe 


where  Ve  is  the  volume  of  solution  in  cubic  centimeters  which 
contains  one  gram-equivalent  of  solute.  The  following  table 
gives  the  specific  and  molar  conductance  of  solutions  of  sodium 
chloride  at  18°  C.:  — 


Concentration. 

Dilution. 

Sp.  Cond. 

Molar  Cond. 

1 

1,000 

0.0744 

74.4 

0.1 

10,000 

0.00925 

92.5 

0.01 

100,000 

0.001028 

102.8 

0.001 

1,000,000 

0.0001078 

107.8 

0.0001 

10,000,000 

0.00001097 

109.7 

It  will  be  observed  that  the  molar  conductance  increases  with  the 
dilution  up  to  a  certain  point  beyond  which  it  remains  nearly 
constant.  That  the  molar  conductance  should  change  but  little 
will  become  apparent  from  the  following  considerations:  — 
Imagine  a  rectangular  cell  of  indefinite  height  and  having  a  cross- 
sectional  area  of  one  square  centimeter,  and  further  assume  that 
two  opposite  walls  can  function  as  electrodes.  Let  1000  cc.  of  a 
solution  containing  one  mol  of  solute  be  introduced  into  the  cell, 
and  let  its  conductance  be  determined.  Now  let  the  solution  be 
diluted  to  2000  cc.  and  the  conductance  of  the  diluted  solution  be 
measured.  While  the  specific  conductance  of  the  diluted  solution 
is  reduced  to  one-half  of  its  original  value,  yet  since  the  electrode 


350 


THEORETICAL  CHEMISTRY 


surface  in  contact  with  the  solution  is  doubled,  owing  to  the  fact 
that  the  solution  stands  at  twice  the  original  height  in  the 
cell,  the  total  conductance  due  to  one  mol  of  solute  remains  un- 
changed. This,  of  course,  is  only  the  case  with  completely  ionized 
solutes. 

Determination  of  Electrical  Conductance.  The  determination 
of  the  electrical  conductance  of  a  solution  resolves  itself  into 
the  determination  of  its  resistance  by  a  simple  modification  of 
the  familiar  Wheatstone-bridge  method.  The  arrangement  of  the 
apparatus  for  this  method  devised  by  Kohlrausch  *  is  represented 
diagrammatically  in  Fig.  84,  where  ab  is  the  bridge  wire,  B  is  a 


Fig.  84. 

resistance  box,  and  C  is  a  cell  containing  the  solution  whose 
resistance  is  to  be  measured.  The  points  d  and  c  are  connected 
to  a  small  induction  coil  /  which  gives  an  alternating  current. 
This  is  necessary  in  order  to  prevent  polarization  which  would 
occur  if  a  direct  current  were  used.  The  use  of  the  alternating 
current  necessitates  the  substitution  of  a  telephone,  T,  for  the 
galvanometer  usually  employed  in  measuring  resistance.  The 
positions  of  the  induction  coil  and  telephone  are  sometimes  inter- 
changed, but  the  arrangement  shown  in  the  diagram  is  to  be  pre- 
ferred, since  it  insures  a  high  electromotive  force  where  the  sliding 
*  Wied.  Ann.,  6,  145  (1879);  n,  653  (1880);  26,  161  (1885). 


ELECTRICAL   CONDUCTANCE 


351 


contact  c  touches  the  wire,  this  being  the  most  uncertain  connec- 
tion in  the  entire  arrangement.  A  small  accumulator  A,  serves 
to  operate  the  induction  coil.  In  making  a  measurement,  the 
coil  is  connected  with  the  accumulator  and  the  vibrator  adjusted 
so  that  a  high  " mosquito-like "  tone  is  emitted;  then  the  sliding 
contact  c  is  moved  along  the  wire  ab  until  the  sound  in  the  tele- 
phone reaches  a  minimum,  the  position  of  the  point  of  contact 
with  the  ^bridge-wire  being  read  on  the  millimeter  scale  placed 
below.  According  to  the  principle  of  the  Wheatstone  bridge,  it 
follows  that 

C  =  bc 

B      ac 


Since  the  resistance  B  and  the  lengths  be  and  ac  are  known,  the 
resistance  C  can  be  calculated.  Various  types  of  conductance 
cells  are  in  use,  depending  upon  whether 
the  solution  has  a  high  or  a  low  resistance. 
The  form  shown  in  Fig.  85  is  widely  used. 
The  two  electrodes  are  made  of  platinum 
foil,  connection  with  the  mercury  hi  the  two 
glass  tubes  it  being  established  by  means  of 
two  pieces  of  stout  platinum  wire  sealed 
through  the  ends  of  these  tubes.  The  tubes  it 
are  fastened  into  a  tight-fitting  vulcanite 
cover  so  that  the  electrodes  may  be  re- 
moved, rinsed  and  dried  without  altering 
their  relative  positions.  Before  the  cell  is 
used,  the  electrodes  must  be  coated  electro- 
lytically  with  platinum  black.  It  is  not 
necessary  to  know  the  area  of  the  elec- 
trodes or  the  distance  between  them,  since 
it  is  possible  to  determine  a  factor,  termed 
the  resistance  capacity,  by  means  of  which 
the  results  obtained  with  the  cell  can  be 
transformed  into  reciprocal  ohms.  To  this 


Fig.  85. 


end   the   specific  conductances  of  a  number  of  standard  solu- 
tions have  been  carefully  determined  by  Kohlrausch;   thus,  for 


352  THEORETICAL  CHEMISTRY 

a  0.02  molar  solution  of  potassium  chloride  he  found  the  following 
values :  — 

*i8°  =  0.002397    and     *2i°  =  0.002768, 
or 

Aiso  =  119.85    and    A25°  =  138.54. 

Let  the  resistance  of  the  cell  when  filled  with  0.02  molar  potassium 
chloride  be  C,  then  according  to  the  principle  of  the  Wheatstone 
bridge  we  have 

r      n  bc 

0    =  £>  •  —  j 

ac 

or  denoting  the  conductance  of  the  solution  by  L,  we  obtain 

T       I         ac 

Ju  —  -7=;  — 


C      B-bc 

Since  the  specific   conductance  K  must  be  proportional  to  the 
observed  conductance,  we  have 


where  K  is  the  resistance  capacity  of  the  cell.     If  the  measure- 
ment is  made  at  18°  C.,  then  we  have 

0.002397  B  •  bc 


K 


ac 


Having  determined  the  resistance  capacity  of  the  cell  we  may 
then  proceed  to  determine  the  conductance  of  any  solution.  For 
example,  suppose  that  when  the  resistance  in  the  box  is  B'}  the 
point  of  balance  on  the  bridge-wire  is  at  c',  then  the  specific  con- 
ductance of  the  solution  will  be 

K'  =  #_E£L 
B'bc' 

If  *'  is  multiplied  by  the  volume  of  the  solution,  we  obtain  the 
equivalent  conductance,  or 

A  =  K'V. 

Relative  Conductances  of  Different  Substances.  The  study 
of  the  electrical  conductance  of  various  solutes  in  aqueous  solu- 
tion, reveals  the  fact  that  electrolytes  differ  greatly  in  their  con- 
ducting power.  They  may  be  roughly  divided  into  two  classes:  — 


ELECTRICAL   CONDUCTANCE 


353 


those  with  high  conducting  power,  such  as  strong  acids,  strong 
bases,  and  salts;  and  those  with  low  conducting  power,  such  as 
ammonia  and  most  of  the  organic  acids  and  bases.  Further- 
more, the  equivalent  or  molar  conductance  increases  with  the  dilu- 
tion until  a  dilution  of  about  10,000  liters  is  reached,  beyond 
which  it  remains  constant.  The  following  table  gives  the  equiv- 
alent conductances  of  three  typical  electrolytes,  V  representing 
the  volume  of  the  solution  in  liters,  and  A  the  equivalent  con- 
ductance :  — 

HYDROCHLORIC    ACID. 


V 

A  (18°). 

0.333 

201.0 

1.0 

278.0 

10.0 

324.4 

100.0 

341.6 

1000.0 

345.5 

SODIUM  HYDROXIDE. 


V 

A  (18°). 

0.333 

100.7 

1.0 

149.0 

10.0 

170.0 

100.0 

187.0 

500.0 

186.0 

POTASSIUM  CHLORIDE. 


V 

A  (18°). 

0.333 

82.7 

1.0 

91.9 

10.0 

104.7 

100.0 

114.7 

1,000.0 

119.3 

10,000.0 

120.9 

354 


THEORETICAL  CHEMISTRY 


The  curves  shown  in  Fig.  86  are  plotted  from  the  data  of  the 
foregoing  table,  and  bring  out  very  clearly  the  differences  in  con- 
ducting power  possessed  by  the  three  electrolytes. 

In  general  the  conductance  of  pure  liquids  is  small.  Thus,  the 
specific  conductance  of  pure  water  at  18°  is  approximately  1  X  10~6 


Dilution,  V 
Fig.  86. 


reciprocal  ohms  and,  as  Walden  *  has  shown,  the  specific  conduct- 
ance of  a  number  of  other  solvents  is  of  the  same  order  as  that 
for  water.  Mixtures  of  two  liquids,  each  of  which  is  practically 
non-conducting,  may  have  a  conductance  differing  but  little  from 
that  of  the  two  components;  or  the  mixture  may  have  a  very 
high  conductance.  For  example,  the  conductance  of  a  mixture 

*  Zeit.  phys.  Chem.,  46,  103  (1903). 


ELECTRICAL   CONDUCTANCE  355 

of  water  and  ethyl  alcohol  is  of  the  same  order  of  magnitude  as 
that  of  the  two  components,  while  on  the  other  hand,  a  mixture 
of  water  and  sulphuric  acid,  each  of  which  in  the  pure  state  is 
practically  a  non-conductor,  has  great  conducting  power.  The 
variation  of  the  specific  conductance  of  mixtures  of  water  and 
sulphuric  acid  is  represented  in  Fig.  87,  the  concentrations  of  sul- 
phuric acid  being  plotted  on  the  axis  of  abscissae  and  the  specific 
conductances  on  the  axis  of  ordinates.  It  appears  that  as  the 


20          30         40          60         60          70         80          90        100         110 
Per  Cent  Sulphuric  Aqjg 

Fig.  87. 

concentration  of  the  sulphuric  acid  increases,  the  specific  conduct- 
ance of  the  mixture  increases  until  30  per  cent  of  acid  is  present, 
beyond  which  point  it  gradually  diminishes.  When  pure  sul- 
phuric acid  is  present  the  value  of  the  specific  conductance  is 
practically  zero.  On  dissolving  sulphur  trioxide  in  the  pure  acid, 
the  specific  conductance  increases  slightly  to  a  maximum  and  then 
falls  rapidly  to  zero.  There  is  a  minimum  in  the  curve  corre- 
sponding to  about  85  per  cent  of  acid,  a  concentration  which 


356 


THEORETICAL  CHEMISTRY 


corresponds  almost  exactly  with  the  hydrate  H2SO4.H2O.  Why 
some  liquid  mixtures  should  have  marked  conducting  power  and 
others  hardly  any,  it  is  difficult  to  explain.  Many  fused  salts, 
such  as  silver  nitrate  and  lithium  chloride,  are  excellent  conductors 
and  are  thus  exceptions  to  the  general  rule,  that  pure  substances 
belonging  to  the  second  class  of  conductors  possess  little  conduct- 
ing power. 

The  Law  of  Kohlrausch.  The  electrical  conductance  of  solu- 
tions was  systematically  investigated  by  Kohlrausch  who  showed 
that  the  limiting  value  of  the  equivalent  conductance,  which  may 
be  represented  by  A«,  is  different  for  different  electrolytes  and 
may  be  considered  as  the  sum  of  two  independent  factors,  one  of 
which  refers  to  the  cation  and  the  other  to  the  anion.  This  experi- 
mental result  is  commonly  known  as  the  law  of  Kohlrausch. 

The  limiting  value  of  the  equivalent  conductance  is  reached 
when  the  molecules  are  completely  broken  down  into  ions,  and 
under  these  conditions  the  whole  of  the  electrolyte  participates 
in  conducting  the  current.  The  accompanying  table,  giving  the 
equivalent  conductances  at  infinite  dilution  of  several  binary 
electrolytes,  illustrates  the  truth  of  the  law  of  Kohlrausch. 


EQUIVALENT  CONDUCTANCES  AT  INFINITE  DILUTION. 


K 

Na 

Li 

NH4 

H 

Ag 

Cl.. 

123 

103 

95 

122 

353 

NO3 

118 

98 

350 

109 

OH 

228 

201 

C1O3 

115 

103 

C2H3O2.         .   . 

94 

73 

83 

The  differences  between  two  corresponding  sets  of  numbers  in 
the  same  vertical  column,  and  of  any  two  corresponding  sets  of 
numbers  in  the  same  horizontal  row  will  be  found  to  be  nearly 
equal.  This  could  only  occur  when  the  limiting  conductance  is 
the  sum  of  two  entirely  independent  quantities.  Each  ion 
invariably  carries  the  same  charge  of  electricity  and  moves  with 


ELECTRICAL   CONDUCTANCE  357 

its  own  velocity  quite  independent  of  the  nature  of  its  compan- 
ion ion.     Therefore,  at  infinite  dilution,  we  have 

A  oo   =   *c  +  la, 

in  which  lc  and  la  are  the  equivalent  conductances  of  the  ions  01 
the  electrolyte  at  infinite  dilution.     From  this  it  follows  that 


Z.      A 
and 

1  —  n  —  —    a  ' 

fcTt     AV 

or 

lc  =  nAoo, 
and 

la  =  (1  —  n)Aoo. 


,  the  equivalent  conductance  of  silver  nitrate  at  infinite  dilu- 
tion at  18°  is  115.5,  while  n  =  0.518  and  1  -  n  =  0.482;  there- 
fore 

lc  =  0.518  X  115.5  =  60.8, 
and 

Z/=  0.482  X  115.5  =  55.7; 

or  one  gram-equivalent  of  silver  ions  possesses  a  conductance  of 
60.8  when  placed  between  electrodes  one  centimeter  apart  and 
large  enough  to  contain  between  them  the  entire  volume  of  solu- 
tion in  which  the  Ag*  ions  exist;  and  one  gram-equivalent  of 
NO3'  ions  under  the  same  conditions  have  a  conductance  equal 
to  55.7. 

The  values  of  the  ionic  conductances  at  infinite  dilution  remain 
constant  in  all  solutions  in  the  same  solvent  at  the  same  temper- 
ature, so  that  it  is  possible  to  calculate  the  equivalent  conductance 
for  any  substance  at  infinite  dilution. 

In  the  subjoined  table  are  given  the  ionic  conductances  of 
various  ions  at  18°  and  infinite  dilution,  together  with  their  temper- 
ature coefficients. 


358  THEORETICAL  CHEMISTRY 

IONIC  CONDUCTANCES  AT  INFINITE  DILUTION. 


Ion. 

l. 

Temp.  Coeff. 

Li* 

33  44 

0  0265 

Na" 

43  55 

0.0244 

K* 

64.67 

0.0217 

Rb* 

67.6 

0.0214 

Cs" 

68.2 

0.0212 

NH4*                       

64.4 

0.0222 

rr  

66.0 

0.0215 

A.gV. 

54.02 

0.0229 

F'.. 

46.64 

0.0238 

Dl' 

65  44 

0.0216 

Br' 

67.63 

0.0215 

[' 

66.40 

0.0213 

3CN' 

56.63 

0.0211 

C1O3'                   

55.03 

0.0215 

[O3'  

33.87 

0.0234 

NXV 

61  78 

0  0205 

ET 

318  0 

DH' 

174  0 

Zn" 

45.6 

0.0251 

Me". 

46.0 

0.0256 

Ba"  ...              

56.3 

0.0238 

Pb"  

61.5 

0.0243 

SO4"  

68.7 

0.0227 

CO3"  

70.0 

0.0270 

In  the  case  of  weak  electrolytes  the  value  of  A*  cannot  be 
determined  directly  from  conductance  measurements,  since  before 
the  limiting  value  is  reached,  the  solution  has  become  so  dilute  as 
to  render  accurate  measurements  of  the  specific  conductance 
impossible.  The  law  of  Kohlrausch  enables  us  to  get  around 
this  difficulty.  Thus,  the  value  of  A«>  for  acetic  acid  must  be 
equal  to  the  sum  of  the  conductances  of  the  H*  and  CH3COO' 
ions.  The  conductance  of  the  H"  ion  at  18°  is,  according  to 
the  preceding  table,  318.  The  value  of  the  conductance  of  the 
CH3COO'  ion  can  be  determined  from  the  conductance  of  sodium 
acetate  at  infinite  dilution,  Aoo  for  this  salt  being  78.1  at  18°. 
Since  the  ionic  conductance  of  the  Na*  ion  is  43.55  at  18°,  it 
follows  that  the  conductance  of  the  CH3COO'  ion  must  be  78.1  — 
43.55  =  34.55.  Therefore,  for  acetic  acid  we  have 

A«  =  lc  +  L  =  318  +  34.55  =  352.55  at  18°. 


ELECTRICAL   CONDUCTANCE 


359 


Bredig  *  has  shown  that  the  ionic  conductance  of  elementary 
ions  is  a  periodic  function  of  the  atomic  weight.     When  the  ionic 


I 

2  80- 

o 


60- 


40- 


20 


40 


80  120 

Atomic  Weight 

'Kg.  88. 


160 


200 


conductances  are  plotted  as  ordinates  against  the  atomic  weights 
as  abscissae,  the  curve  shown  in  Fig.  88  is  obtained.  A  glance  at 
the  curve  shows  the  periodic  nature  of  the  relation. 

Absolute  Velocity  of  the  Ions.  Thus  far  we  have  considered 
only  the  relative  velocities  of  the  ions  and  their  conductances; 
we  now  proceed  to  the  consideration  of  their  absolute  velocities 
in  centimeters  per  second. 

Let  a  current  of  electricity  pass  through  a  centimeter  cube  of  a 
solution  of  a  binary  electrolyte.  If  the  solution  contains  m  mols 
of  solute  per  liter,  then  ra/1000  will  be  the  number  of  mols  in  the 
centimeter  cube.  The  charge  on  either  the  cation  or  the  anion 

,  where  F  =  96,540  coulombs.    If  C  represents  the  total 


isFiooo 

current,  we  have 


C  = 


m 


1000 


(lc  +  la) 


Zeit.  phys.  Chem.,  13,  242  (1894). 


360  THEORETICAL  CHEMISTRY 

since  the  current  is  the  charge  which  passes  through  one  face  of 
the  cube  in  one  second.  In  a  centimeter  cube,  the  current  is 
equal  to  the  product  of  the  specific  conductance  and  the  difference 
of  potential  E,  the  latter  being  numerically  equal  to  the  potential 
gradient,  the  distance  between  the  electrodes  being  one  centi- 
meter. Hence,  we  have 

1000  KE  =  Fm  (lc  +  la). 

If  E  is  expressed  in  volts  and  K  in  reciprocal  ohms,  lc  and  la  will  be 
expressed  in  centimeters  per  second,  for  on  passing  to  absolute 
electromagnetic  units,  we  have 

1000  (K  X  10-9)  (E  X  108)  _       /7    ,  ,  x 
(F  X  10-1)  -wft  +  W, 

or 

1000         f      M 

--  K  =  P      lc  +  la      = 


where  lc  and  la  are  the  ionic  velocities  for  unit  potential  gradient  — 
1  volt  per  centimeter. 
From  this  it  follows  that 


The  equivalent  conductance  of  a  0.0001  molar  solution  of  potas- 
sium chloride  at  18°  is  128.9;  the  total  velocity  of  the  two  ions 
is  then, 

128  9 

0.001345  cm.  per  sec. 


This  total  velocity  is  made  up  of  the  two  individual  ionic  velocities. 
The  transport  numbers  of  the  two  ions,  K*  and  Cl',  are  respectively 
0.493  and  0.507.  Hence  the  absolute  velocities  of  the  ions,  ex- 
pressed in  centimeters  per  second,  in  a  0.0001  molar  solution  of 
potassium  chloride  at  18°  are  as  follows:  — 

u  =  0.001345  X  0.493  =  0.00066  cm.  per  sec., 
and 

v  =  0.001345  X  0.507  =  0.00068  cm.  per  sec0 


ELECTRICAL   CONDUCTANCE 


361 


The  absolute  velocities  of  some  of  the  more  common  ions  at  18° 
are  given  in  the  following  table:  — 

ABSOLUTE   IONIC  VELOCITIES. 


Ion. 

Velocity. 

Ion. 

Velocity. 

K' 

cm.  per  sec. 
0  00066 

H'.. 

cm.  per  sec. 

0.00320 

NH4* 

0  00066 

cr  

0.00069 

Na* 

0  00045 

NO3'  

0.00064 

Li' 

0  00036 

CKY  

0.00057 

Ag* 

0  00057 

OH'.. 

0.00181 

CriO?" 

0  000473 

Cu" 

0.00031 

The  velocities  of  certain  ions  have  been  determined  directly. 
Thus,  the  velocity  of  the  hydrogen  ion  was  measured  by  Lodge  * 
in  the  following  manner:  —  The  tube  B,  Fig.  89,  40  cm.  long  and 
8  cm.  in  diameter,  was  graduated  and  bent  at  right  angles  at  the 


Fig.  89. 

ends.  This  was  filled  with  an  aqueous  solution  of  sodium  chloride 
in  gelatine,  colored  red  by  the  addition  of  an  alkaline  solution  of 
phenolphthalein.  When  the  contents  of  the  tube  had  gelatinized, 
the  latter  was  placed  horizontally,  connecting  two  beakers  filled 
with  dilute  sulphuric  acid  as  shown  in  the  diagram.  A  current 
of  electricity  was  passed  from  one  electrode  A  to  the  other  elec- 
trode C. 

The  hydrogen  ions  from  the  anode  vessel  were  thus  carried  along 
the  tube,  and  discharged  the  red  color  of  the  phenolphthalein  as 
they  migrated  toward  the  cathode.  In  this  manner  the  velocity 

*  Brit.  Assoc.  Report,  p.  393  (1886). 


362  THEORETICAL  CHEMISTRY 

of  the  hydrogen  could  be  observed  under  a  known  potential  gra- 
dient. The  observed  and  calculated  values  agree  excellently.  It 
was  shown  that  the  velocity  of  the  hydrogen  ions  suffered  almost 
no  retardation  from  the  high  viscosity  of  the  gelatine  solution. 
Whetham,*  in  his  experiments  on  ionic  velocity,  employed  two 
solutions  one  of  which  possessed  a  colored  ion,  the  progress  of  the 
latter  being  observed  and  its  velocity  determined  under  unit 
potential  gradient.  For  example,  consider  the  boundary  line 
between  two  equally  dense  solutions  of  the  electrolytes  AC  and 
BO,  C  being  a  colorless  and  A  a  colored  ion.  When  a  current 
passes  through  the  boundary  between  the  two  electrolytes,  the 
anion  C  will  migrate  toward  the  positive  electrode  while  the  two 
cations,  A  and  B,  will  migrate  toward  the  negative  electrode 
and  the  color  boundary  will  move  with  the  current,  its  speed  being 
equal  to  that  of  the  colored  ion  A .  In  this  way  Whetham  measured 
the  absolute  velocities  of  the  ions,  Cu",  Cr207",  and  CY.  Ionic 
velocities  have  also  been  determined  by  Steele  f  who  observed 
the  change  in  the  index  of  refraction  of  the  solution  as  the  ions 
migrated.  The  accompanying  table  gives  a  comparison  of  the 
calculated  and  observed  velocities  of  some  of  the  ions. 


Ion. 

Velocity  (obs.). 

Velocity  (calc.). 

H*.. 

cm.  per  sec. 
0.0026 

cm.  per  sec. 
0  0032 

Cu" 

0  0029 

0  0031 

cr 

0  00058 

0  00069 

Cr2O7" 

0.00047 

0  000473 

Conductance  and  lonization.  We  have  already  seen  that 
solutions  of  strong  acids,  strong  bases  and  salts  exert  abnormally- 
great  osmotic  pressures.  According  to  the  molecular  theory,  this 
abnormal  osmotic  activity  has  been  ascribed  to  the  presence  in 
the  solutions  of  a  greater  number  of  dissolved  particles  than  would 
be  anticipated  from  the  simple  molecular  formulas  of  the  solutes. 
The  ratio  of  the  observed  to  the  theoretical  osmotic  pressure  was 
represented,  according  to  Van't  Hoff,  by  the  factor  "i." 

*  Phil.  Trans.  A.,  184,  337  (1893);  196,  507  (1895). 
f  Phil.  Trans.  A.,  198,  105  (1902). 


ELECTRICAL   CONDUCTANCE  363 

In  1887,  Arrhenius  showed  that  there  is  an  intimate  connection 
between  electrical  conductance  and  abnormal  osmotic  activity, 
only  those  solutions  conducting  the  electric  current  which  exert 
abnormally-high  osmotic  pressures.  It  had  already  been  pointed 
out  by  Kohlrausch,  that  the  equivalent  conductance  of  a  solution 
increases  at  first  with  the  dilution  and  then  ultimately  becomes 
constant.  Arrhenius  explained  this  behavior  by  assuming  that 
the  molecules  of  the  solute  are  dissociated  into  ions,  the  con- 
ductance of  the  solution  being  solely  dependent  upon  the  number 
of  ions  present.  The  dissociation  increases  with  the  dilution  until 
finally,  when  the  equivalent  conductance  has  reached  its  maximum 
value,  it  is  complete,  the  molecules  of  solute  being  entirely  broken 
down  into  ions.  This  theory  of  Arrhenius,  known  as  the  theory 
of  electrolytic  dissociation,  is  based,  as  has  been  pointed  out, 
upon  the  views  advanced  by  Clausius.  Arrhenius  showed  how 
the  degree  of  dissociation  of  an  electrolyte  can  be  calculated 
from  the  electrical  conductance  of  its  solutions.  According  to  the 
theory  of  electrolytic  dissociation,  the  conductance  of  a  solution 
is  dependent  upon  the  number  of  ions  present  in  the  solution, 
upon  their  charges,  and  upon  their  velocities.  Since  the  electric 
charges  carried  by  equivalent  amounts  of  the  ions  of  different 
electrolytes  are  equal,  and  since  the  velocities  of  the  ions  for  the 
same  electrolyte  are  practically  independent  of  the  dilution  of  the 
solution,  it  follows  that  the  increase  in  equivalent  conductance 
with  dilution  must  depend  almost  wholly  upon  the  increase  in  the 
number  of  ions  present. 

The  equivalent  conductance  at  infinite  dilution  has  been  shown 
by  the  law  of  Kohlrausch  to  be 

Aoo    =   lc  +  la, 

and,  therefore,  the  equivalent  conductance  at  any  dilution  v,  must 
be 

Av  =  a  (lc  -f  Za), 

where  a  is  the  degree  of  dissociation  of  the  electrolyte.     Dividing 
the  second  equation  by  the  first,  we  obtain 

' 


364 


THEORETICAL  CHEMISTRY 


This  equation  enables  us  to  calculate  the  degree  of  ionization  of 
an  electrolyte  at  any  dilution,  provided  the  conductance  of  the 
solution  at  the  particular  dilution  is  known,  together  with  its 
conductance,  at  infinite  dilution.  For  example,  Av  at  18°  for  a 
molar  solution  of  sodium  chloride  is  74.3,  and  A«>  is  110.3;  there- 
fore, a  =  74.3  -T-  110.3  =  0.673,  or  in  a  molar  solution,  the  mole- 
cules of  sodium  chloride  are  dissociated  to  the  extent  of  67.3  per 
cent.  A  comparison  of  the  values  of  i  based  upon  conductance 
and  osmotic  data  has  already  been  given  in  the  table  on  page  208. 
Since  Av  =  a.  (lc  +  la),  we  may  also  write 


The  Dissociation  of  Water.     Water  behaves  as  a  very  weak 
binary  electrolyte,  dissociating  according  to  the  equation, 


The  specific  conductance  of  water,  purified  with  the  utmost  care, 
has  been  determined  by  Kohlrausch  and  Heydweiller.*  Their 
results  are  given  in  the  following  table:  — 


Temperature, 
degrees. 

Specific  Conduct- 
anceXlO-e. 

0 

0.014 

18 

0.040 

25 

0.055 

34 

0.084 

50 

0.170 

The  conductance  of  pure  water  at  0°  is  so  small  that  one  milli- 
meter of  it  has  a  resistance  equal  to  that  of  a  copper  wire  of  the 
same  cross-section  and  40,000,000  kilometers  in  length,  or  in 
other  words,  long  enough  to  encircle  the  earth  one  thousand  times. 
Knowing  the  specific  conductance  of  water,  its  degree  of  dissoci- 
ation can  be  easily  calculated.  The  ionic  conductances  of  the 
two  ions  of  water  at  18°  are  as  follows:  —  H*  =  318,  and  OH'  = 
*  Zeit.  phys.  Chem.,  14,  317  (1894). 


ELECTRICAL   CONDUCTANCE  365 

174.     Therefore,  the  maximum  equivalent  conductance  of  water 

should  be 

Aoo  =  318  +  174  =  492. 

The  equivalent  conductance  at  18°,  of  a  liter  of  water  between 
electrodes  1  cm.  apart  is,  according  to  the  data  of  Kohlrausch, 

0.04  X  10~6  X  103  =  0.04  X  10~3; 
therefore 

0.04  X  10-3 


492 


=  0.8  X  10~7  =  c,  the  concentration  of  the  ions, 


H*  and  OH',  in  mols  per  liter  at  18°. 

Conductance  of  Difficultly-Soluble  Salts.  In  a  saturated 
solution  of  a  difficultly-soluble  salt,  the  solution  is  so  dilute  that  in 
general  we  may  assume  complete  ionization,  or  Av  =  Aoo . 

When  this  is  the  case,  we  have 

^solution  —  *H2O  =  K> 

and 

Av  =  Aoo  =  1000  /c7. 
Hence 


1000  K 

or  if  m  denotes  the  concentration  in  gram-equivalents  per  liter, 

we  have 

1       1000  K 


Thus,  Bottger  found  for  a  saturated  solution  of  silver  chloride  at 
20°,  *'  =  1.374  X  10~6.  Deducting  the  specific  conductance  of 
the  water  at  this.  temperature,  we  have 

K  =  1.374  X  10~6  -  0.044  X  10~6  =  1.33  X  10~6. 

Since  the  value  of  Aoo,  at  20°,  for  silver  chloride,  determined  from 
the  table  of  ionic  conductances,  is  125.5,  we  have 


m  =  100°  X13  =  1-06  X  10-'  gr.-equiv.  AgCl  per  liter. 


366 


THEORETICAL  CHEMISTRY 


Temperature  Coefficient  of  Conductance.  When  the  temper- 
ature of  a  solution  of  an  electrolyte  is  raised,  the  equivalent  con- 
ductance usually  increases.  The  increase  in  conductance  is  due, 
not  to  an  increase  in  the  ionization,  but  to  the  greater  velocity 
of  the  ions  caused  by  the  diminution  of  the  viscosity  of  the  solution. 
According  to  Kohlrausch,  the  relation  between  conductance  and 
temperature  may  be  approximately  expressed  by  the  following 
equation, 


where  /3  is  the  temperature  coefficient,  or  change  in  conductance 
for  1°  C.     Solving  the  equation  for  0,  we  have 

A,  -  A18o 


iso  (t  -  18) 

The  temperature  coefficients  of  several  of  the  more  common  elec- 
trolytes are  given  in  the  accompanying  table. 


TEMPERATURE  COEFFICIENTS  OF  CONDUCTANCE. 


Electrolyte. 


Nitric  acid 

Sulphuric  acid 

Hydrochloric  acid 

Potassium  hydroxide 

Potassium  nitrate 

Potassium  iodide 

Potassium  bromide 

Potassium  chlorate 

Silver  nitrate 

Potassium  chloride 

Ammonium  chloride 

Potassium  sulphate 

Copper  sulphate 

Sodium  chloride 

Sodium  sulphate 

Zinc  sulphate 


Temperature 
Coefficient. 


0.0163 
0.0164 
0.0165 
0.0190 
0.0211 
0.0212 
0.0216 
0.0216 
0.0216 
0.0217 
0.0219 
0.0223 
0.0225 
0.0226 
0.0234 
0.0250 


The  temperature  coefficient  of  conductance  is  not,  however,  a 
simple  linear  function  of  the  temperature.     The  following  empiri- 


ELECTRICAL   CONDUCTANCE  367 

cal  equations,  expressing  equivalent  conductance  at  infinite  dilu- 
tion at  any  temperature  t  in  terms  of  the  conductance  at  18°,  have 
been  derived  by  Kohlrausch :  — 

A*,*  =  Aooiso }  1  +  a  (t  -  18)  +  ft  (t  -  18)2}, 
and 

ft  =  0.0163  (a  -  0.0174). 

When  the  values  of  Ac»i8°,  a,  and  /?,  as  determined  for  a  large 
number  of  electrolytes,  are  substituted  in  the  above  equation,  he 
showed  that  Aoof  becomes  equal  to  zero  at  a  temperature  approx- 
imating to  —  40°.  Kohlrausch  suggested  that  each  ion  moving 
through  the  solution  carries  with  it  an  "atmosphere"  of  solvent, 
and  that  the  resistance  offered  to  the  motion  of  the  ion  is  simply 
the  frictiohal  resistance  between  masses  of  pure  water.  This 
view  is  in  harmony  with  the  solvate  theory  discussed  in  an  earlier 
chapter.  Washburn  *  has  calculated  the  degree  of  ionic  hydration 
for  several  ions.  He  finds,  for  example,  that  the  hydrogen  ion 
carries  with  it  0.3  molecule  of  water,  while  the  lithium  ion  is 
hydrated  to  the  extent  of  4.7  molecules  of  water. 

Conductance  at  High  Temperatures  and  Pressures.  The 
conductance  of  several  typical  electrolytes,  at  temperatures  rang- 
ing from  that  of  the  room  up  to  306°,  have  been  measured  by  A.  A. 
Noyes  and  his  co-workers.f  These  determinations  were  made  in 
a  conductance  cell  especially  constructed  to  withstand  high 
pressures. 

The  results  show  that  the  values  of  A<»  for  binary  electrolytes 
become  more  nearly  equal  with  rise  of  temperature.  This  may 
be  taken  as  an  indication  of  the  fact  that  the  ionic  velocities  tend 
to  become  more  nearly  equal  as  the  temperature  rises.  The 
conductance  of  ternary  electrolytes  increases  uniformly  with  the 
temperature,  and  attains  values  which  are  considerably  greater 
than  those  reached  by  binary  electrolytes.  This  is  what  might 
be  expected,  since  if  an  ion  is  bivalent,  as  in  a  ternary  electrolyte, 
the  driving  force  is  greater,  and  the  ion  must  move  faster,  and, 
consequently,  the  conductance  must  be  greater. 

*  Jour.  Am.  Chem.  Soc.,  30,  322  (1909). 

t  Publication  of  Carnegie  Institution,  No.  63. 


368  THEORETICAL  CHEMISTRY 

The  temperature  coefficient  of  conductance  for  binary  elec- 
trolytes is  greater  between  100°  and  156°,  than  below  or  above 
these  temperatures.  The  temperature  coefficients  of  ternary 
electrolytes  increases  uniformly  with  rising  temperature.  In  the 
case  of  acids  and  bases,  the  rate  of  increase  in  conductance  steadily 
diminishes  as  the  temperature  rises.  The  ionization  decreases 
regularly  with  rise  in  temperature,  the  temperature  coefficient  of 
ionization  being  small  between  18°  and  100°.  The  effect  of  pres- 
sure on  conductance  was  studied  by  Fanjung.*  He  found  that 
the  conductance  increases  slightly  with  increasing  pressure. 
This  result  he  interprets  as  being  due  to  increased  ionic  velocity 
rather  than  to  an  increase  in  the  number  of  ions  present  in  the 
solution. 

Conductance  of  Non-aqueous  Solutions.  A  large  amount  of 
interesting  and  important  work  has  been  done  in  recent  years 
upon  the  electrical  conductance  of  solutions  in  non-aqueous  sol- 
vents. 

It  is  impossible  to  give  even  a  brief  survey  of  the  results  of  these 
investigations,  and  we  must  limit  ourselves  to  the  statement  of 
the  following  general  conclusions :  — f 

(1)  The  conditions  in  non-aqueous  solutions  are  much  more 
complex  than  in  aqueous  solutions. 

(2)  In  general,  the  laws  which  have  been  found  to  apply  to  aque- 
ous solutions  also  apply  to  non-aqueous  solutions. 

(3)  Different   solvents  appear  to  have  different  dissociating 
powers. 

(4)  The  dissociating  power  appears  to  run  parallel  with  the 
dielectric  constant  of  the  solvent. 

Many  interesting  phenomena  present  themselves  in  connec- 
tion with  the  conductance  of  electrolytes  in  mixed  solvents,  but 
for  an  account  of  this  work  the  student  must  consult  the  original 
papers  of  Jones  and  his  students.  | 

*  Zeit.  phys.  Chem.,  14,  673  (1894). 

t  "  Elektrochemie  der  nichtwassrigen  Losungen,"  by  G.  Carrara,  Ahren'a 
"Sammlung  Chemischer  und  chemisch-technischer  Vortraege,"  Vol.  XII. 
J  Publication  of  the  Carnegie  Institution,  No.  80. 


ELECTRICAL   CONDUCTANCE 


369 


Ionizing  Power  of  Solvents.  Thomson  *  and  Nernst  f  pointed 
out  that  if  the  forces  which  hold  the  atoms  in  the  molecule  are  of 
electrical  origin,  then  those  liquids  which  possess  large  dielectric 
constants  should  have  correspondingly  great  ionizing  power. 
This  is  a  direct  consequence  of  Coulomb's  law  of  electrostatic 
attraction,  which  may  be  expressed  by  the  equation, 


"  Kd?' 

in  which  q\  and  $2  denote  two  electric  charges,  d  the  distance 
between  them,  /  the  force  of  attraction  and  K  the  dielectric  con- 
stant. Obviously  the  larger  K  becomes,  the  smaller  will  be  the 
value  of  /;  i.  e.,  the  more  likely  the  molecule  will  be  to  break  down 
into  ions.  That  the  above  relation  is  approximately  true  may  be 
seen  from  the  following  table  :  — 

DIELECTRIC  CONSTANTS. 


Solvent. 

K 

Ionizing  Power. 

B6nZ6I16 

2.3 

Extremely  weak 

Ethyl  ether     

4.1 

Weak 

Ethyl  alcohol                                           

25 

Fairly  strong 

Formic  acid                                                  

62 

Strong 

Water                                                     

80 

Very  strong 

Hydrocyanic  acid                          

96 

Very  strong 

Dutoit  and  Aston  t  have  suggested  that  there  is  a  connection 
between  the  ionizing  power  of  a  solvent  and  its  degree  of  associa- 
tion, and  Dutoit  and  Friderich  §  conclude  that  the  values  of  A«> , 
for  a  given  electrolyte  dissolved  in  different  solvents,  are  a  direct 
function  of  the  degree  of  association  and  an  inverse  function  of 
the  viscosity  of  the  solvents.  Water  and  the  alcohols  furnish 
good  illustrations  of  the  truth  of  this  generalization. 

*  Phil.  Mag.,  36,  320  (1893). 

t  Zeit.  phys.  Chem.,  13,  531  (1894). 

t  Compt.  rend.,  125,  240  (1897). 

§  Bull.  Soc.  Chim.  [3],  19,  321  (1898). 


370 


THEORETICAL  CHEMISTRY 


Conductance  of  Fused  Salts.  While  solid  salts  are  exceed- 
ingly poor  conductors  of  electricity,  yet  as  the  temperature  is 
raised  their  conductance  increases  until  at  their  melting  point 
they  may  be  grouped  with  good  conductors.  There  is  no  sudden 
increase  in  conductance  at  the  melting  point.  The  specific 
conductance  of  a  fused  salt  may  exceed  the  specific  conductance 
of  the  most  concentrated  aqueous  solutions,  but  owing  to  the  high 
concentration  the  equivalent  conductance  is  much  less.  The 
following  table  gives  the  specific  and  equivalent  conductance  of 
fused  silver  nitrate :  — 


Temperature, 
degrees. 

Sp.  Cond. 

Equiv.  Cond. 

218  (melt,  pt.) 

0.681 

29.2 

250 

0.834 

36.1 

300 

1.049 

46.2 

350 

1.245 

55.4 

The  specific  conductance  of  a  60  per  cent  aqueous  solution  of 
silver  nitrate  at  18°  is  0.208  reciprocal  ohms. 

If  the  salts  are  impure  the  conductance  is  raised,  the  effect  of 
impurities  being  apparent  even  before  the  salts  have  reached  their 
melting  points.  This  is  analogous  to  the  behavior  of  solutions, 
and  suggests  that  the  impurity  functions  in  the  salt  mixture  as  a 
dissolved  solute.* 

PROBLEMS. 

1.  An  aqueous  solution  of  copper  sulphate  is  electrolyzed  between 
copper  electrodes  until  0.2294  gram  of  copper  is  deposited.     Before  elec- 
trolysis the  solution  at  the  anode  contained  1.1950  grams  of  copper,  after 
electrolysis  1.3600  grams.     Calculate  the  transport  numbers  of  the  two 
ions,  Cu"  and  SO/'.  Ans.  n  =  0.28,  1  -  n  =  0.72. 

2.  A  solution  containing  0.1605  per  cent  of  NaOH  was  electrolyzed 
between   platinum   electrodes.     After'  electrolysis   55.25   grams   of  the 
cathode  solution  contained  0.09473  gram  of  NaOH,  whilst  the  concen- 
tration of  the  middle  portion  of  the  electrolyte  was  unchanged.    In  a 

*  For  a  complete  treatment  of  fused  electrolytes  the  student  is  advised  to 
consult,  "Die  Elektrolyse  geschmolzener  Salze,"  by  Richard  Lorenz. 


ELECTRICAL   CONDUCTANCE  371 

silver  coulometer  the  equivalent  of  0.0290  gram  of  NaOH  was  deposited 
during  electrolysis.  Calculate  the  transport  numbers  of  the  Na"  and  OH' 
ions.  Ans.  n  =  0.791,  1  -  n  =  0.209. 

3.  A  0.02  molar  solution  of  potassium  chloride  gives  in  a  certain  con- 
ductance cell  a  resistance  of  150  ohms.     Find  the  resistance  capacity  of 
the  cell  at  18°  and  25°. 

4.  In  a  0.01  molar  solution  of  potassium  nitrate,  the  transport  num- 
bers of  the  cation  and  anion  are,  respectively,  0.503  and  0.497.     Find  the 
equivalent  conductances  of  the  two  ions  in  this  solution  having  given  that 
its  specific  conductance  is  0.001044.  Ans.  lc  =  52.5,  la  =  51.9. 

5.  The  absolute  velocity  of  the  Ag*  ion  is  0.00057  cm.  per  sec.,  and  that 
of  the  Cl'  ion  is  0.00059  cm.  per  sec.      Calculate  the  equivalent  con- 
ductance of  an  infinitely  dilute  solution  of  silver  chloride. 

6.  The  equivalent  conductance  of  an  infinitely  dilute  solution  of  am- 
monium chloride  is  130;  the  ionic  conductances  of  the  ions  OH/  and  Cl' 
are  174  and  65.44  respectively.     Calculate  the  equivalent  conductance 
of  ammonium  hydroxide  at  infinite  dilution.  Ans.   Aoo  =  238.56. 

7.  The  equivalent  conductance  of  a  molar  solution  of  sodium  nitrate 
at  18°  is  66;   its  conductance  at  infinite  dilution  is  105.3.     What  is  the 
degree  of  ionization  in  the  molar  solution?        Ans.  a  =  62.6  per  cent. 

8.  The  specific  conductance  of  a  saturated  solution  of  AgCN  at  20° 
is  1.79  X  10~6  and  the  specific  conductance  of  water  at  the  same  temper- 
ature is  0.044  X  10~6  reciprocal  ohms.     The  equivalent  conductance  at 
infinite  dilution  is  115.5.     Calculate  the  solubility  of  AgCN  in  grams  per 
liter.  Ans.   2.2  X  10~4  gram. 

9.  The  equivalent  conductance  at  18°  of  a  solution  of  sodium  sulphate 
containing  0.1  gram-equivalent  of  salt  per  liter  is  78.4,  the  conductance 
at  infinite  dilution  is  113  reciprocal  ohms.     What  is  the  value  of  i  for 
the  solution?    What  is  its  osmotic  pressure? 

Ans.  i  =  2.388;  osmotic  pressure  =  5.7  atmos. 

10.  The  freezing-point  of  a  0.1  molar  solution  of  CaCl2  is  —  0°.482. 
(a)  Calculate  the  degree  of  ionization  (freezing  point  constant  =  1.89 
for  one  mol  per  liter),     (b)  Calculate  the  degree  of  ionization  from  the 
equivalent  conductance  at  18°,  which  is  82.79  reciprocal  ohms,  whilst  the 
equivalent  conductance  of  CaC^  at  infinite  dilution  is  115.8  reciprocal 
ohms.  Ans.   (a)  a  =  0.755;     (b)  a  =  0.715. 


CHAPTER  XVI. 
ELECTROLYTIC  EQUILIBRIUM  AND  HYDROLYSIS. 

Ostwald's  Dilution  Law.  It  has  been  shown  in  preceding 
chapters  that  the  law  of  mass  action  is  applicable  to  chemical 
equilibria  in  both  gaseous  and  liquid  systems.  We  now  proceed 
to  show  that  it  applies  equally  to  electrolytic  equilibria.  When 
acetic  acid  is  dissolved  in  water  it  dissociates  according  to  the 
equation 

CHaCOOH  ?±  CH3COO'  +  IT. 

Let  one  mol  of  acetic  acid  be  dissolved  in  water  and  the  solution 
diluted  to  v  liters,  and  let  a  denote  the  degree  of  dissociation. 

Then,  the  concentration  of  the  undissociated  acid  is  —  —  ,  and 

the  concentration  of  the  ions  is  -  •    Applying  the  law  of  mass 
action,  we  have 


or 


where  K  is  the  equilibrium  or  ionization  constant. 

This  equation  expressing  the  relation  between  the  degree  of 
ionization  and  dilution,  was  derived  by  Ostwald*  and  is  known 

as  the  Ostwald  dilution  law.     Since  a  =  — — ,  we  may  substitute 

Aoo 

this  value  of  a  in  equation  (1)  and  obtain  the  expression 

—!L K  (2) 

Aoo  (A^  -  A,)  v 

*  Zeit.  phys.  Chem.,  2,  36  (1888);  3,  170  (1889). 
372 


ELECTROLYTIC  EQUILIBRIUM  AND  HYDROLYSIS      373 

The  dilution  law  may  be  tested  by  substituting  the  value  of  a, 
corresponding  to  any  dilution  v,  in  the  equation  and  calculating 
the  value  of  the  ionization  constant,  K\  the  value  of  a  at  any 
other  dilution  may  then  be  calculated  and  compared  with  the 
value  determinedly  direct  experiment.  The  following  table  gives 
the  results  obtained  with  acetic  acid  at  14°.l,  K  being  equal  to 
0.0000178:  — 


v  (in  liters). 

oXlO2  (calc.). 

crXlO2  (obs.). 

0.994 

0.42 

0.40 

2.02 

0.60 

0.614 

15.9 

1.67 

1.66 

18.1 

1.78 

1.78 

1,500.0 

15.0 

14.7 

3,010.0 

20.2 

20.5 

7,480.0 

30.5 

30.1 

15,000.0 

40.1 

40.8 

As  will  be  seen,  the  agreement  between  the  observed  and  cal- 
culated values  is  very  close.  The  table  also  shows  to  how  small 
an  extent  the  molecules  of  acetic  acid  are  broken  down  into  ions, 
a  molar  solution  being  dissociated  less  than  0.5  per  cent.  The 
dilution  law  holds  for  nearly  all  organic  acids  and  bases,  but  fails 
to  apply  to  salts,  strong  acids,  and  strong  bases.  When  a  is 
small,  the  term  (1  —  a)  does  not  differ  appreciably  from  unity, 
and  equation  (1)  becomes 


or 


(3) 


On  the  other  hand,  when  a  cannot  be  neglected,  we  have,  on 
solving  equation  (2)  for  a, 

..  V  I  ..9T^9. 

(4) 


The   method    of  derivation  indicates  that  the  dilution  law  is 
only  strictly   applicable    to    binary  electrolytes   and,  therefore, 


374  THEORETICAL  CHEMISTRY 

it  is  improbable  that  it  will  hold  for  electrolytes  yielding  more 
than  two  ions.  It  has  been  found,  however,  that  organic  acids 
whether  they  are  mono-,  di-,  or  polybasic  always  ionize  as 
a  monobasic  acid  up  to  the  dilution  at  which  a  =  50  per 
cent.  This  means  that  the  dilution  law  is  applicable  to  poly- 
basic  acids  up  to  that  dilution  at  which  the  acid  is  50  per  cent 
ionized. 

Strength  of  Acids  and  Bases.  There  are  several  methods  by 
which  the  relative  strengths  of  acids  can  be  estimated.  A  method 
which  has  proved  of  great  value  is  that  in  which  two  different 
acids  are  allowed  to  compete  for  a  certain  base,  the  amount  of 
which  is  insufficient  to  saturate  both  of  them.  Suppose  equiva- 
lent weights  of  nitric  and  dichloracetic  acids  together  with  sufficient 
potassium  hydroxide  to  saturate  one  acid  completely  are  taken: 
we  then  determine  the  position  of  the  equilibrium  represented  by 
the  equation 

HN03  +  CHC12  •  COOK  <=>  CHC12  •  COOH  +  KN03. 

In  order  to  determine  the  conditions  of  equilibrium  we  may  make 
use  of  any  method  which  does  not  disturb  this  equilibrium.  Since 
ordinary  chemical  methods  are  excluded  on  this  account,  we 
employ  any  physical  property  which  is  capable  of  exact  measure- 
ment and  differs  sufficiently  in  the  two  systems,  as  for  example, 
the  change  in  volume,  or  the  thermal  change,  accompanying 
neutralization.  Thus,  Ostwald*  found  that  when  one  mol  of 
potassium  hydroxide  is  neutralized  by  nitric  acid  in  dilute  solu- 
tion, the  volume  increases  approximately  20  cc.  When  one  mol 
of  potassium  hydroxide  is  neutralized  by  dichloracetic  acid,  how- 
ever, the  increase  in  volume  is  13  cc.  It  is  evident,  therefore,  that 
if  nitric  acid  completely  displaces  dichloracetic  acid  as  represented 
by  the  above  equation,  the  increase  in  volume  will  be  20  —  13 
=  7  cc.;  if  no  displacement  occurs,  then  the  volume  will  remain 
constant.  He  found  that  the  volume  actually  increased  5.67  cc. 
Therefore,  the  reaction  represented  by  the  upper  arrow  has  pro- 
ceeded to  the  extent  of  5.67  -j-  7  =  80  per  cent.  That  is  to  say, 

*  Jour,  prakt.  Chem.  [2],  18,  328  (1878). 


ELECTROLYTIC  EQUILIBRIUM   AND  HYDROLYSIS      375 

in  the  competition  of  the  two  acids  for  the  base,  the  nitric  acid 
has  taken  80  per  cent  and  the  dichloracetic  acid  has  taken  20  per 
cent,  or  the  relative  strengths  of  the  two  acids  are  in  the  ratio  of 
80  :  20,  or  4  :  1. 

The  relative  strengths  of  acids  can  also  be  determined  from  their 
catalytic  effect  on  the  rates  of  certain  reactions,  such  as  the 
hydrolysis  of  esters  or  the  inversion  of  cane  sugar. 

The  order  of  the  activity  of  acids  is  the  same  whether  measured 
by  equilibrium  or  kinetic  methods.  Arrhenius  pointed  out  that 
the  relative  strengths  of  acids  can  be  readily  determined  from 
their  electrical  conductance.  The  order  of  the  strengths  of  acids 
as  determined  by  equilibrium  and  kinetic  methods  is  the  same  as 
that  of  their  electrical  conductances  in  equivalent  solutions. 
This  is  well  illustrated  by  the  following  table  in  which  the  three 
methods  are  compared,  hydrochloric  acid  being  taken  as  the 
standard  of  comparison :  — 


Acid. 

Method  Employed. 

Equilibrium. 

Kinetic. 

Conductance. 

HC1 

100 
100 

49 
9 

100 
100 

53.6 
4.8 
0.4 

100 
99.6 
65.1 
4.8 
1.4 

HNO3.  . 

H2SO4. 

CH2C1COOH  

CH3COOH  

The  results  of  these  and  other  experiments  warrant  the  con- 
clusion that  the  strength  of  an  acid  is  determined  by  the  number 
of  hydrogen  ions  which  it  yields.  It  is  important  to  note  that  the 
electrical  conductance  of  an  acid  is  not  directly  proportional  to 
its  hydrogen  ion  concentration;  the  relatively  high  velocity  of 
the  H  ion  is  the  cause  of  the  approximate  proportionality  between 
these  two  variables.  In  the  case  of  a  weak  acid,  the  value  of  the 
ionization  constant  may  be  taken  as  a  measure  of  the  strength  of 
the  acid.  The  following  table  gives  the  values  of  the  ionization 
constants  at  25°  for  several  different  acids. 


376 


THEORETICAL  CHEMISTRY 
IONIZATION  [CONSTANTS  OF  ACIDS. 


Acid. 

lonization 
Constant. 

Acetic  acid  

0.0000180 

Monochloracetic  acid  
Trichloracetic  acid 

0.00155 
1  21 

Cyanacetic  acid 

0  0037 

Formic  acid  

0.000214 

Carbonic  acid  

3040  XlO-10 

Hydrocyanic  acid  

570  XlO-10 

Hydrogen  sulphide  

13  XlO-10 

Phenol 

1  3  XlO~10 

Since  for  a  weak  acid,  a  =  \^vK,  it  follows  that  for  two  weak 
acids  at  the  same  dilution,  we  may  write 


-=vf> 

«  *    A. 


or  the  ratio  of  the  degrees  of  ionization  of  the  two  acids  is  equal  to 
the  square  root  of  the  ratio  of  their  ionization  constants.  Thus, 
from  the  data  given  in  the  foregoing  table  for  acetic  and  mono- 
chloracetic  acids,  we  have 

K000018       1 


0.00155       9.3 

or  the  effect  of  replacing  one  atom  of  hydrogen  in  the  methyl 
group  of  acetic  acid  increases  the  strength  of  the  acid  about  nine 
times. 

Just  as  the  hydrogen  ion  concentration  of  acids  determines  their 
strength,  so  the  strength  of  bases  is  determined  by  the  concen- 
tration of  hydroxyl  ions.  The  strength  of  bases  may  be  estimated 
by  methods  similar  to  those  employed  in  determining  the  strength 
of  acids.  Thus,  two  different  bases  may  be  allowed  to  compete 
for  an  amount  of  acid  sufficient  to  saturate  only  one  of  them;  or 
a  catalytic  method  developed  by  Koelichen  *  may  be  used.  This 
method  is  based  upon  the  effect  of  hydroxyl  ions  on  the  rate  of 
condensation  of  acetone  to  diacetonyl  alcohol,  as  represented  by 
the  equation 

2CH3COCH3  =  CH3COCH2C  (CH3)2OH. 
*  Zeit.  phya.  Chem.,  33,  129  (1900). 


ELECTROLYTIC  EQUILIBRIUM   AND  HYDROLYSIS      377 

In  addition  to  these  two  methods,  the  method  of  electrical  con- 
ductance is  also  applicable.  The  agreement  between  the  results 
obtained  by  the  three  methods  is  quite  satisfactory.  The  alkali 
and  alkaline  earth  hydroxides  are  very  strong  bases  and  are  dis- 
sociated to  about  the  same  extent  as  equivalent  solutions  of 
hydrochloric  and  nitric  acids,  while  on  the  other  hand,  ammonia 
and  many  of  the  organic  bases  are  very  weak.  The  following 
table  gives  the  ionization  constants  of  several  typical  bases :  — 


IONIZATION  CONSTANTS  OF  BASES. 


Base. 

Ionization 
Constant. 

Ammonia 

0  000023 

Methylamine  .  .  . 

0  00050 

Trimethylamine  .... 

0  000074 

Pyridine  .  .                        . 

2.5X10-10 

Aniline  

1  .  1  X  10-10 

Mixtures  of  Two  Electrolytes  with  a  Common  Ion.  Just  as 
the  dissociation  of  a  gaseous  substance  is  diminished  by  the  addi- 
tion of  an  excess  of  one  of  the  products  of  dissociation,  so  the 
ionization  of  weak  acids  and  bases  is  depressed  by  the  addition  of 
a  salt  with  an  ion  common  to  the  acid  or  the  base.  If  the  degree 
of  ionization  of  a  salt  with  an  ion  in  common  with  an  acid  or  a 
base  is  represented  by  a',  and  n  denotes  the  number  of  molecules 
of  salt  present,  then  the  equation  of  equilibrium  of  the  acid  or 
base  will  be 

(naf  +  a)a  =  Kv(l-  a), 

where  a  is  the  degree  of  ionization  of  the  acid  or  base.  For  very 
weak  acids  and  bases,  a  is  so  small  that  1  —  a  does  not  differ 
appreciably  from  unity,  and  since  a'  is  practically  independent  of 
the  dilution,  we  obtain 

na  =  Kv 


or 


a  = 


Kv 
n 


378  THEORETICAL  CHEMISTRY 

That  is,  the  ionization  of  a  weak  acid  or  base,  in  the  presence  of 
one  of  its  salts,  is  approximately  inversely  proportional  to  the 
amount  of  salt  present. 

In  many  of  the  processes  of  analytical  chemistry,  advantage  is 
taken  of  the  action  of  neutral  salts  on  the  ionization  of  weak  acids 
and  bases.  Thus,  while  the  concentration  of  hydroxyl  ions  in 
ammonium  hydroxide  is  sufficient  to  precipitate  magnesium  hy- 
droxide from  solutions  of  magnesium  salts,  the  presence  of  a  small 
amount  of  ammonium  chloride  depresses  the  ionization  of  the 
ammonium  hydroxide  to  such  an  extent  that  precipitation  no 
longer  takes  place. 

Isohydric  Solutions.  Arrhenius  *  was  the  first  to  point  out 
what  relation  must  exist  between  solutions  of  two  electrolytes 
with  a  common  ion,  in  order  that,  when  mixed  in  any  proportions, 
they  may  not  exert  any  mutual  influence.  He  showed  that  when 
the  concentration  of  the  common  ion  in  each  of  the  two  solutions 
is  the  same  before  mixing,  no  alteration  in  the  degree  of  ionization 
will  occur  after  mixing.  Such  solutions  are  said  to  be  isohydric. 
Thus,  an  aqueous  solution  containing  one  mol  of  acetic  acid  in 
8  liters,  is  isohydric  with  an  aqueous  solution  containing  one  mol 
of  hydrochloric  acid  in  667  liters.  On  mixing  these  two  solutions 
the  hydrogen  ion  concentration  remains  unchanged,  and  if  the 
mixture  is  treated  with  a  small  amount  of  sodium  hydroxide, 
equal  amounts  of  sodium  acetate  and  sodium  chloride  will  be 
formed. 

That  isohydric  solutions  may  be  mixed  without  altering  their 
respective  ionizations  may  be  shown  in  the  following  manner:  — 
Let  C  and  c  denote  the  concentrations  of  the  undissociated  por- 
tions, and  CA,  C%,  CA,  and  c2  denote  the  concentrations  of  the  dis- 
sociated portions  of  two  electrolytes,  and  let  Cz  and  <%  correspond 
to  two  different  ions. 

Then,  we  have 

kc  =  cAd,  (1) 

and 

KC  =  CAC2.  (2) 

*  Wied.  Ann.,  30,  51  (1887). 


ELECTROLYTIC  EQUILIBRIUM  AND  HYDROLYSIS      379 

If  v  liters  of  the  first  solution  be  mixed  with  V  liters  of  the  second 
solution,  the  concentrations  of  the  undissociated  portions,  and  of 
the  dissimilar  ions,  will  be 

Cv  cv  C%V  ctf) 

V~+~v'     V  +  v'    V^Tv  T+V 

while  the  concentration  of  the  common  ion  A,  becomes 

CAV  +  CAV 

Applying  the  law  of  mass  action,  we  have 

kc  =     Ay  . — —  Czj  (3) 

and 

CAV  +  cAv~  ... 


But  equations  (3)  and  (4)  only  become  identical  with  equations  (1) 
and  (2)  when  CA  =  CA,  or  in  other  words,  no  change  in  the  degree 
of  dissociation  takes  place  after  the  two  solutions  are  mixed. 

lonization  of  Strong  Electrolytes.  It  has  already  been  men- 
tioned that  the  Ostwald  dilution  law,  which  is  a  direct  conse- 
quence of  the  law  of  mass  action,  applies  only  to  weak  electrolytes. 
Just  why  the  law  of  mass  action  should  fail  to  apply  to  strong 
electrolytes  is  not  known,  but  several  possible  causes  have  been 
suggested  to  account  for  its  failure.  One  of  the  most  plausible 
explanations  is  that  advanced  by  Biltz,*  who  attributes  the 
failure  of  the  law  of  mass  action  when  applied  to  strong  electrolytes, 
to  hydration  of  the  solute.  If  the  ions  become  associated  with 
a  large  proportion  of  the  solvent,  the  effective  ionic  concentration 
would  then  be  the  ratio  of  the  amount  of  the  ion  present  to  that 
of  the  free  solvent,  instead  of  to  the  total  solvent,  as  ordinarily 
calculated.  This  view  is  in  harmony  with  certain  facts  which 
have  been  adduced  in  favor  of  the  theory  of  solvation.  While 
the  Ostwald  dilution  law  does  not  apply  to  strongly  ionized  elec- 
trolytes, certain  empirical  expressions  have  been  derived  which 
*  Zeit.  phys.  Chem.,  40,  218  (1902). 


380 


THEORETICAL  CHEMISTRY 


hold  fairly  well  over  a  wide  range  of  dilution.     Thus,  Rudolphi  * 
showed  that  the  equation 


_   _        K, 

(1  -a)Vv~ 

gives  approximately  constant  values  for  Kr  for  strong  electrolytes. 
The  following  table  gives  the  results  obtained  with  solutions  of 
silver  nitrate  at  25°;  the  numbers  in  the  third  column  being  cal- 
culated by  means  of  the  Ostwald  dilution  law,  while  those  in  the 
fourth  column  are  calculated  by  means  of  Rudolphi's  dilution  law. 


V 

a 

K 

K' 

16 

0.8283 

0.253 

1.11 

32 

0.8748 

0.191 

1.16 

64 

0.8993 

0.127 

1.06 

128 

0.9262 

0.122 

1.07 

256 

0.9467 

0.124 

1.08 

512 

0.9619 

0.125 

1.09 

The  Rudolphi  equation  was  modified  by  Van't  Hoff  f  to  the 
form 


K,, 


This  equation  holds  even  more  closely  than  that  of  Rudolphi. 
A  generalized  form  of  the  Van't  Hoff  equation  has  been  proposed 
by  Bancroft  t  as  follows  :  — 

K"' 


(1  -  a)Vv 

in  which  the  constant  K"'  and  n  are  functions  of  the  nature  of 
the  electrolyte.  Bancroft  suggests  that  this  relation  may  be  of 
the  form 

n  =  2-f, 

where  the  constant  /  varies  between  0  and  a  value  approximating 
0.5.  For  potassium  chloride  at  18°,  if  n  be  placed  equal  to  1.36, 

*  Ibid,  17,  385  (1895). 

t  Zeit.  phys.  Chem.,  18,  300  (1895). 

J  Ibid.,  31,  188  (1899). 


ELECTROLYTIC  EQUILIBRIUM  AND  HYDROLYSIS      381 

K"'  =  2.63.     The  accompanying  table  gives  a  comparison  of  the 
calculated  values  of  a  with  those  obtained  by  direct  experiment. 


V 

a  (obs.)- 

a  (calc.). 

0.3 

0.673 

0.672 

0.5 

0.706 

0.700 

1.0 

0.748 

0.745 

2.0 

0.780 

0.786 

5.0 

0.821 

0.834 

10.0 

0.853 

0.864 

20.0 

0.883 

0.892 

50.0 

0.915 

0.917 

100.0 

0.934 

0.934 

200.0 

0.950 

0.948 

500.0 

0.965 

0.962 

1,000.0 

0.973 

0.970 

2,000.0 

0.978 

0.976 

5,000.0 

0.984 

0.983 

10,000.0 

0.987 

0.987 

Heat  of  lonization.     The  heat  of  ionization  of  an  electrolyte 
can  be  calculated  by  means  of  the  reaction  isochore  equation  of 
Van't  Hoff  (see  p.  274),  provided  the  degree  of  ionization  at  two 
different  temperatures  is  known. 
Since 


and 


it  follows  that  the  heat  of  ionization  may  be  calculated  by  means 
of  the  equation 


Q 


Arrhenius  *  has  shown  that  this  equation  also  applies  to  those 
electrolytes  which  do  not  obey  the  Ostwald  dilution  law.     Some 


Zeit.  phys.  Chem.,  4,  96,  1889. 


382 


THEORETICAL  CHEMISTRY 


of  the  results  obtained  by  Arrhenius  are  given  in  the  accompany- 
ing table :  — 


Electrolyte. 

Temperature. 

Calories. 

Acetic  acid                                                              j 

35° 

386 

Propionic  acid  .                                     .                     5 

21°.  5 
35° 

-28 
557 

Butyric  acid     ....           .  .        J 

21°.  5 
35° 

183 
935 

Phosphoric  acid  5 

21°.  5 
35° 

427 
2458 

Hydrochloric  acid 

21°.  5 
35° 

2103 
1080 

Potassium  chloride 

35° 

362 

Potassium  bromide  

35° 

425 

Potassium  iodide  

35° 

916 

Sodium  chloride  

35° 

454 

Sodium  hydroxide  

35° 

1292 

Sodium  acetate 

35° 

391 

It  will  be  found  that  the  values  of  the  heats  of  ionization  given 
in  this  table  do  not  agree  with  the  values  calculated  for  these 
same  substances  from  the  data  given  in  the  table  on  page  257. 
The  reason  for  this  lack  of  agreement  is,  that  the  data  of  the  earlier 
table  refer  to  the  heat  of  formation  of  the  ions  from  the  dissolved 
substance,  whereas  the  data  of  the  table  just  given  represent  the 
combined  thermal  effects  of  solution  and  ionization. 

The  Solubility  Product.  While  the  law  of  mass  action  does 
not  in  general  apply  to  the  equilibrium  between  the  dissociated 
and  undissociated  portions  of  an  electrolyte,  —  except  in  the  case 
of  organic  acids  and  bases,  —  it  does  apply  with  a  fair  degree  of 
accuracy  to  saturated  solutions  of  electrolytes. 

A  saturated  solution  of  silver  chloride  affords  an  example  of 
such  an  equilibrium.  This  salt  is  practically  completely  ionized 
in  a  saturated  solution,  as  represented  by  the  equation, 


Applying  the  law  of  mass  action  to  this  equilibrium,  we  obtain 

CAg»  X  CGI  _  j^ 

CAgCl 


ELECTROLYTIC  EQUILIBRIUM  AND  HYDROLYSIS      383 

Since  the  solution  is  saturated,  the  value  of  CASCI  must  remain 
constant  at  constant  temperature,  and  therefore 

CAg«  X  CCK  =  constant  =  s, 

where  the  product  of  the  ionic  concentrations  s,  is  called  the  solu- 
bility or  ionic  product. 

The  equilibria  in  the  above  heterogeneous  system  may  be  repre- 
sented thus  :  — 

Ag  +Cl'?±AgCl<=>AgCL 

(in  solution)       (Solid) 

The  solubility  product  for  silver  chloride  at  25°  is  1.56  X  10~10, 
the  ionic  concentrations  being  expressed  in  mols  per  liter.  Hence, 
since  the  two  ions  are  present  in  equivalent  amounts,  a  saturated 
solution  of  silver  chloride  at  25°  must  contain  Vl.56  X  10~10 
=  1.25  X  10~5  mols  per  liter  of  Ag*  and  Cl'  ions.  In  general,  if 

nA  +±  niAi  + 


represents  the  equilibrium  between  an  electrolyte  and  its  products 
of  dissociation  in  saturated  solution,  we  have 


The  solubility  product  may  be  defined  as  the  maximum  product  of 
the  ionic  concentrations  of  an  electrolyte  which  can  exist  at  any  one 
temperature. 

Just  as  the  dissociation  of  a  gaseous  substance  or  of  an  organic 
acid  is  depressed  by  the  addition  of  one  of  the  products  of  dis- 
sociation, so  when  a  substance  with  a  common  ion  is  added  to  the 
saturated  solution  of  an  electrolyte,  the  dissociation  is  depressed 
and  the  undissociated  substance  is  precipitated. 

The  following  example  will  serve  to  illustrate  how  the  solu- 
bility product  of  a  substance  can  be  determined,  and  how  the 
change  in  solubility  due  to  the  addition  of  a  substance  containing 
a  common  ion  may  be  calculated.  The  solubility  of  silver  bromate 
at  25°  is  0.0081  mol  per  liter.  If  we  assume  complete  ionization, 
the  concentration  of  the  ions,  Ag*  and  Br(V  will  be  the  same 
and  equal  to  0.0081  mol  per  liter,  or 

(0.0081)  (0.0081)  =  s. 


384  THEORETICAL  CHEMISTRY 

The  solubility  in  a  solution  of  silver  nitrate  containing  0.1  mol 
of  Ag"  ions  can  be  calculated  from  the  equation, 

(0.0081)2  =  (0.0081  +  0.1-0;)  (0.0081  -  x), 

where  x  represents  the  amount  of  silver  bromate  thrown  out  of  so- 
lution by  the  addition  of  0.1  mol  of  Ag*  ion.  Since  (0.0081  —  x) 
represents  the  concentration  of  Ag"  and  BrO'fc  ions  after  the 
addition  of  the  silver  nitrate,  it  also  represents  the  solubility  of 
silver  bromate  under  similar  conditions.  The  effect  of  adding  a 
solution  of  a  soluble  bromate  containing  0.1  mol  of  BrO3'  ion  will 
be  the  same  as  that  produced  by  0.1  mol  of  Ag"  ion. 

The  Basicity  of  Organic  Acids.  The  Ostwald  dilution  law 
holds  strictly  for  all  monobasic  organic  acids,  and  also  for  poly- 
basic  organic  acids  which  are  less  than  50  per  cent  ionized.  The 
neutral  salts  of  these  acids,  however,  are  much  more  highly  ionized, 
and  the  difference  in  conductance  between  two  dilutions  of  a  neu- 
tral salt  of  a  polybasic  acid  is  greater  than  the  difference  in 
conductance  between  the  same  dilutions  of  a  neutral  salt  of 
a  monobasic  acid.  Ostwald  *  has  shown  that  it  is  possible 
to  estimate  the  basicity  of  an  organic  acid  from  the  difference 
in  the  equivalent  conductance  of  its  sodium  salt  at  two  different 
dilutions. 

As  the  result  of  a  long  series  of  experiments,  he  found  that  the 
difference  between  the  equivalent  conductance  of  the  sodium  salt 
of  a  monobasic  organic  acid  at  v  =  32  liters  and  at  v  =  1024  liters 
is  approximately  10  units.  Similarly,  the  difference  for  a  dibasic 
acid  between  the  same  dilutions  is  20  units,  and  for  an  n-basic 
acid  the  difference  is  10  ft.  Hence,  to  estimate  the  basicity  of  an 
organic  acid,  the  equivalent  conductance  of  its  sodium  salt  at  v  = 
32  liters  and  at  v  =  1024  liters  is  determined;  then,  if  A  is  the  dif- 
ference between  the  values  of  the  conductance  at  the  two  dilutions, 

the  basicity  will  be  n  =  JQ  • 

The  following  table  gives  the  values  of  A  and  n  for  the  sodium 
salts  of  several  typical  organic  acids :  — 

*  Zeit.  phys.  Chem.,  i,  105  (1887);  2,  902  (1888). 


ELECTROLYTIC  EQUILIBRIUM  AND  HYDROLYSIS       385 


Acid. 

A 

• 

Formic  .             

10.3 

1 

Acetic       

9.5 

1 

Propionic 

10  2 

1 

Benzoic 

8  3 

1 

Quininic 

19  8 

2 

Pyridine-tricarboxylic  (1,  2,  3) 

31  0 

3 

Pyridine-tricarboxylic  (1,  2,  4)  

29  4 

3 

Pyridine-tetracarboxylic  

41.8 

4 

Pyridine-pentacarboxylic  

50.1 

5 

Influence  of  Substitution  on  lonization.  Attention  has  already 
been  called  to  the  marked  difference  in  the  strength  of  acetic 
acid  produced  by  the  replacement  of  the  hydrogen  atoms  of  the 
methyl  group  by  chlorine.  In  the  accompanying  table  the  ioniza- 
tion  constants  for  various  substitution  products  of  acetic  acid 
are  given :  — 


Acid. 


lonization 
Constant  (25°). 


Acetic  CH3COOH                                   

0  000018 

Propionic,  CH3CH2COOH 

0.000013 

Chloracetic,  CH2C1COOH      

0.00155 

Bromacetic,  CH2BrCOOH  

0.00138 

Cyanacetic,  CH2CNCOOH  

0.00370 

Glycollic,  CH2OHCOOH  

0.000152 

Phenylacetic,  C6H5CH2COOH  

0.000056 

Amidoacetic,  CH2NH2COOH  

3.4X10-10 

This  table  affords  an  interesting  illustration  of  the  influence  of 
different  substituents  on  the  strength  of  acetic  acid.  Thus,  the 
activity  of  the  acid  is  increased  by  the  replacement  of  alkyl  hydro- 
gen atoms  by  Cl,  Br,  CN,  OH,  or  CeH5,  while  the  substitution  of 
the  CH3  or  NH2  groups  diminishes  its  activity.  If  we  assume 
that  the  substituents  retain  their  ion-forming  capacity  on  enter- 
ing into  the  molecule  of  acetic  acid,  these  differences  in  activity 
can  be  readily  explained.  Thus,  Cl,  Br,  CN,  and  OH  tend  to 
form  negative  ions,  and  hence  increase  the  negative  character  of 
the  group  into  which  they  enter.  On  the  other  hand,  basic  groups, 


386  THEORETICAL  CHEMISTRY 

such  as  NH2,  diminish  the  tendency  of  the  group  into  which  they 
enter  to  yield  negative  ions. 

The  influence  of  an  alkyl  residue  on  the  strength  of  an  organic 
acid  is  conditioned  by  its  distance  from  the  carboxyl  group.  This 
is  well  illustrated  by  the  ionization  constants  of  propionic  acid 
and  some  of  its  derivatives. 


Acid. 

Ionization 
Constant  (25°). 

Propionic  acid,  CH3CH2COOH  

0  0000134 

Lactic  acid,  CH3CHOHCOOH  

0  000138 

/8-oxypropionic  acid,  CH2OHCH2COOH 

0  0000311 

The  effect  of  the  OH  group  in  the  a-position  is  seen  to  be  much 
more  marked  than  when  it  occupies  the  /3-position. 

The  position  of  a  substituent  in  the  benzene  nucleus  exerts  a 
marked  influence  on  the  strength  of  the  derivatives  of  benzoic 
acid.  The  ionization  constants  of  benzoic  acid  and  the  three 
chlorbenzoic  acids  are  given  in  the  following  table :  — 


Acid. 

Ionization 
Constant  (25°). 

Benzoic  acid,  C6H6COOH 

0  000073 

o-Chlorbenzoic  acid,  C6H4C1COOH 

0  00132 

m-Chlorbenzoic  acid,  C6H4C1COOH 

0  000155 

p-Chlorbenzoic  acid,  C6H4C1COOH  

0  000093 

When  the  halogen  enters  the  ortho-position,  the  strength  of  the 
acid  is  greatly  augmented,  while  in  the  meta-  and  para-  positions 
the  effect  is  much  smaller,  meta-chlorbenzoic  acid  being  stronger 
than  para-chlorbenzoic  acid.  It  is  a  general  rule  that  the  influence 
of  substituents  is  always  greatest  in  the  ortho-position,  and  least 
in  the  meta-  and  para-  positions,  the  order  in  the  two  latter  being 
uncertain. 

Hydrolysis.  When  a  salt  formed  by  a  weak  acid  and  a  strong 
base,  such  as  sodium  carbonate,  is  dissolved  in  water,  the  solution 


ELECTROLYTIC  EQUILIBRIUM  AND  HYDROLYSIS      387 

shows  an  alkaline  reaction,  while  on  the  other  hand,  when  a  salt 
formed  by  a  strong  acid  and  a  weak  base,  such  as  ferric  chloride, 
is  dissolved  in  water,  the  solution  shows  an  acid  reaction. 

The  process  which  takes  place  in  the  aqueous  solution  of  a  salt 
causing  it  to  react  alkaline  or  acid,  is  termed  hydrolysis  or  hydro- 
lytic  dissociation.  If  MA  represents  a  salt,  in  which  M  is  the  basic 
and  A  is  the  acidic  portion,  then  the  hydrolytic  equilibrium  may 
be  represented  by  the  equation 

MA  +  H20  ?±  MOH  +  HA. 

If  the  base  formed  is  insoluble  or  undissociated  and  the  acid  is 
dissociated,  the  solution  will  react  acid.  If  the  acid  formed  is 
insoluble  or  undissociated  and  the  base  is  dissociated,  the  solu- 
tion will  react  alkaline.  Finally,  if  both  base  and  acid  are  insoluble 
or  undissociated,  the  salt  will  be  completely  transformed  into  base 
and  acid,  and,  as  there  will  be  no  excess  of  either  H*  or  OH'  ions, 
the  solution  will  remain  neutral. 

It  is  evident,  then,  that  hydrolysis  is  due  to  the  removal  of 
either  one  or  both  of  the  ions  of  water  by  the  ions  of  the  salt  to 
form  undissociated  or  insoluble  substances.  As  fast  as  the  ions 
of  water  are  removed,  the  loss  is  made  good  by  the  dissociation  of 
more  water,  until  eventually  a  condition  of  equilibrium  is  estab- 
lished. The  conditions  governing  hydrolytic  equilibrium  may  be 
determined  from  a  knowledge  of  the  solubility  or  ionic  constant 
of  the  substances  involved.  Thus,  if  the  product  of  the  concen- 
trations of  the  ions  M*  and  OH'  exceeds  that  which  can  exist  in 
pure  water,  then  some  undissociated  or  insoluble  substance  will 
be  formed.  This  will  disturb  the  equilibrium  of  H*  and  OH' 
ions,  and  a  further  dissociation  of  water  must  occur  until  the 
ionic  product  of  water  is  just  reached. 

If  now  the  ions  H"  and  A'  do  not  unite  to  form  undissociated 
acid,  the  presence  of  an  excess  of  H*  ions  will  disturb  the  equi- 
librium between  pure  water  and  its  products  of  dissociation;  or, 
since 


CH«  X 
the  concentration  of  OH'  ions  present,  when  CH«  represents  the 

total  concentration  of  H*  ions,  will  be  ^^  • 

CH- 


388  THEORETICAL  CHEMISTRY 

A  similar  readjustment  will  take  place  when  an  undissociated 
or  insoluble  acid  and  a  dissociated  base  are  formed. 

We  may  now  proceed  to  consider  three  different  cases  of  hydroly- 
sis, viz.,  when  the  reaction  is  caused  (1)  by  the  base,  (2)  by  the 
acid,  and  (3)  by  both  base  and  acid. 

CASE  1.  The  formation  of  an  undissociated  or  insoluble  base  is 
primarily  the  cause  of  the  hydrolysis,  the  acid  formed  being  dis- 
sociated. 

Let  the  hydrolytic  equilibrium  be  represented  by  the  equation 


The  reaction  will  proceed  in  the  direction  of  the  upper  arrow  until 
the  product,  CM*  X  COH',  exceeds  that  which  can  exist  in  the  ab- 
sence of  an  undissociated  base.  When  equilibrium  is  established, 
we  have 

final  CM-  X  final  COH'  =  -^MOH  X  CMOH  formed,  (1) 

or  if  the  base  formed  is  practically  insoluble,  the  equilibrium  equa- 
tion simplifies  to  the  form 

final  CM*  X  final  COH'  =  SMOH,  (2) 

where  SMOH  is  the  solubility  product  of  the  base.  The  condition 
of  equilibrium  represented  by  the  equation 

CH-  X  COH'  =  SH2o, 

must  be  fulfilled.  It  follows  that  the  final  concentration  of  the 
OH'  ions  will  be  the  quotient  obtained  by  dividing  the  ionic  product 
for  water,  at  the  temperature  of  the  experiment,  by  the  final  con- 
centration of  the  H"  ion,  this  latter  being  wholly  dependent  upon 
the  extent  of  the  reaction  and  the  degree  of  ionization  of  the  acid 
formed.  If  the  degree  of  hydrolysis  of  the  salt  be  represented  by 
x,  and  the  degree  of  dissociation  of  the  unhydrolyzed  portion  of 
the  salt  be  denoted  by  a,,  then,  if  one  mol  of  salt  be  dissolved 
in  V  liters  of  solution,  the  final  concentration  of  M"  ions  will  be 

a*    v  —  i  and  the  final  concentration  of  the  undissociated  base 

/y»  *>* 

will  be-    The  total  acid  formed  will  be      ,  and  if  aa  denotes 


ELECTROLYTIC  EQUILIBRIUM   AND  HYDROLYSIS      389 


the  degree  of  dissociation  of  the  acid,  the  concentration  of  the  H* 
ions  will  be  aa 
(2),  we  obtain 


JM 

ions  will  be  aa  --    Substituting  these  values  in  equations  (1)  and 


a*  (1  ~  x)     SH,O       ~  Z 

-JT~  3  =  ^MOH  X  pi  (3) 

and 


Simplifying  equations  (3)  and  (4),  we  have 

&        t  Q?0  =    gnto    = 
(1  —  a:)  F    ««      ^MOH 
and 


From  equations  (5)  and  (6)  it  appears  that  the  constant  of  hydrol-- 
ysis  can  be  found  from  the  ionic  product  for  water  and  either  the 
ionization  constant  or  the  solubility  product  of  the  base  which 
causes  the  hydrolysis.  Furthermore,  if  the  base  formed  is  insol- 
uble, equation  (6)  shows  that  the  degree  of  hydrolysis,  x,  is  inde- 
pendent of  the  dilution  of  the  salt,  V. 

CASE  II.  The  formation  of  an  undissociated  or  insoluble  add 
is  primarily  the  cause  of  the  hydrolysis,  the  base  formed  being  dis- 
sociated. In  this  case  hydrolysis  takes  place  until  the  product 
CH*  X  CA'  exceeds  that  which  can  exist  in  the  absence  of  undis- 
sociated acid.  When  equilibrium  is  established,  we  have 

final  cH-  X  final  CA'  =  KRA  X  CHA  formed,  (7) 

or  if  the  acid  formed  is  practically  insoluble,  the  equilibrium  equa- 
tion simplifies  to  the  form 

final  CH«  X  final  CA'  =  SHA-  (8) 


390  THEORETICAL  CHEMISTRY 

Since  the  final  CH«  =  SH,O  •=-  final  COET,  we  have,  final  CA*  =   *    v — - ' 

/y« 

final  COH'=  aby,  where  ab  is  the  degree  of  dissociation  of  the  base 

/v» 

formed,  and  the  final  CHA  =  y  •     Substituting  these  values  in  equa- 
tions (7)  and  (8),  we  obtain 

a,  (1  —  x) 


,     ab,       <T 

and 

«.  (1  -  x) 


Simplifying  equations  (9)  and  (10),  we  have 


and 

(T^)-S  =  Sf  =  ^'-  (12) 

It  is  evident  from  equations  (11)  and  (12),  that  the  constant  of 
hydrolysis  can  be  found  from  the  ionic  product  for  water  and 
either  the  ionization  constant  or  the  solubility  product  of  the  acid 
which  causes  the  hydrolysis. 

CASE  III.  The  formation  of  an  acid  and  a  base,  both  being 
slightly  dissociated,  is  the  cause  of  the  hydrolysis. 

In  this  case  let  us  assume  that  Kn\  is  smaller  than 


Since  the  final  CQH' =  ^MOH  X  CMOH  ,  and  since  both  HA  and 

CM- 

MOH  are  slightly  dissociated,  we  may  write  CHA  =  CMOH  =  ^ ,  and 

a.  (1  ~  X) 
CA*  =  CM*  = f7 * 


ELECTROLYTIC  EQUILIBRIUM   AND  HYDROLYSIS      391 


Substituting  these  values  in  equation  (7),  we  obtain 

X 


,       . 

(16) 


a,  (1  ~  X) 
V 

Simplifying  equation  (13),  we  obtain 
x* 


.      =  = 

(1-x)   c? 


From  equation  (14)  we  see  that  the  constant  of  hydrolysis  can  be 
found  from  the  ionic  product  and  the  ionization  constants  of  the 
acid  and  the  base.  If  both  acid  and  base  are  practically  insolu- 
ble, the  reaction  will  be  complete  at  all  dilutions. 

As  an  illustration  of  the  application  of  the  foregoing  equations, 
we  may  take  the  calculation  of  the  degree  of  hydrolytic  dissociation 
of  potassium  cyanide  in  0.1  molar  solution  at  25°.  Potassium 
cyanide  being  a  salt  of  a  weak  acid,  the  degree  of  hydrolysis  can 
be  calculated  by  means  of  the  equation 


(\-x)V  a. 
Since  at  25°,  #HA  =  13  X  1Q-10  and  sH2o  =  (0.91  X  10~7),  we  have 

SHZO  =  (0.91  X  IP"7) 
#HA        13  X  10-10 

and  since  in  dilute  solution  at  =  <*&  =  1,  we  have 

re2          =  (0.91  X  IP"7)2 
(1  -  x)  •  10         13  X  10-10 
or 

x  =  0.00798. 

Experimental  Determination  of  Hydrolysis.  The  degree  of 
hydrolysis  can  be  determined  experimentally  in  several  different 
ways.  A  very  convenient  method  is  that  based  upon  measure- 
ments of  electrical  conductance.  When  a  salt  reacts  hydrolyti- 
cally  with  one  mol  of  water,  the  limiting  value  of  its  equivalent 


392  THEORETICAL  CHEMISTRY 

conductance  will  be  A  A  +  AB,  where  A  A  and  A  5  denote  the  equiv- 
alent conductances  of  the  acid  and  base  formed.  If  A  is  the  equiv- 
alent conductance  of  the  unhydrolyzed  salt,  and  AA  is  the  actual 
conductance  of  the  salt  at  the  same  dilution,  then  the  increase  in 
conductance  corresponding  to  a  degree  of  hydrolysis  x,  will  be 
AA  —  A.  The  value  of  A  may  be  found  by  determining  the 
conductance  of  the  salt  in  the  presence  of  an  excess  of  one  of  the 
products  of  hydrolysis  and  deducting  from  it  the  conductance  of 
the  substance  added.  Since  if  the  hydrolysis  were  complete,  the 
equivalent  conductance  would  beAA+A^  —  A,  we  have 

Aft  -A 

' 


all  conductances  being  measured  at  the  same  dilution  and  the 
same  temperature.  The  following  example  will  illustrate  the 
use  of  this  equation  :  —  At  25°,  the  equivalent  conductance  of  an 
aqueous  solution  of  aniline  hydrochloride  is  118.6,  the  dilution 
being  99.2  liters.  The  equivalent  conductance  in  the  presence  of 
an  excess  of  aniline  is  103.6,  while  the  equivalent  conductance  of 
hydrochloric  acid  at  the  same  dilution  is  411.  The  conductance 
of  pure  aniline  is  so  small  as  to  be  negligible.  Substituting  these 
values  in  the  equation,  we  find 

118.6  -  103.6 
:    411  -  103.6   : 

Lunden  *  has  shown  how  this  method  may  be  extended  to  cases 
where  both  acid  and  base  are  slightly  dissociated. 

The  lonization  Constant  of  Water.  One  of  the  most  accurate 
methods  known  for  the  determination  of  the  ionization  constant 
of  water  is  based  upon  measurements  of  the  degree  of  hydrolytic 
dissociation  of  different  salts.  Thus  Shields  f  found  that  a  0.1 
molar  solution  of  sodium  acetate  is  0.008  per  cent  hydrolyzed  at 
25°.  We  may  consider  the  salt,  as  well  as  the  sodium  hydroxide 
formed  from  its  hydrolysis,  to  be  completely  dissociated  at  this 
dilution.  The  ionization  constant  of  the  acetic  acid  formed  is 

*  Jour.  chim.  phys.,  5,^145,  574  (1907). 
t  Zeit.  phys.  Chem.,  12,  167  (1893). 


ELECTROLYTIC  EQUILIBRIUM   AND  HYDROLYSIS       393 

0.000018  at  25°.     Solving  equation  (11)  (on  page  390)  for  SH,O, 
and  remembering  that  at  =  <*&  =  1,  we  have 


SH,O  = 


Substituting  the  above  values  in  this  expression,  we  obtain 


SH,o  =  0.000018 


10 


"  1M  X 


and  since  the  ions,  H*  and  OH',  are  present  in  equivalent  amounts, 
we  have 


CH-  =  COH'  =  v  1.16  X  10~14  =  1.1  X  10~7  mol  per  liter. 


Kohlrausch  obtained  from  his  measurements  of  the  conductance 
of  pure  water  at  25°,  CH«  =  COH'  =  1.05  X  10~7  mol  per  liter  (see 
p.  365). 

PROBLEMS. 

1.  At  25°  the  specific  conductance  of  butyric  acid  at  a  dilution  of 
64  liters  is  1.812  X  10~4  reciprocal  ohms.    The  equivalent  conductance 
at  infinite  dilution  is  380  reciprocal  ohms.     What  is  the  degree  of  ioniza- 
tion  and  the  concentration  of  H*  ions  in  the  solution?    What  is  the  ioni- 
zation  constant  of  the  acid? 

Ans.  CL  =  0.0305,  c  H-  =  4.765  X  10~4  mol  per  liter,  K  =  1.5  X  10~5. 

2.  The  heat  of  neutralization  of  nitric  acid  by  sodium  hydroxide  is 
13,680  calories,  and  of  dichloracetic  acid,  14,830  calories.  "  When  one 
equivalent  of  sodium  hydroxide  is  added  to  a  dilute  solution  containing 
one  equivalent  of  nitric  acid  and  one  equivalent  of  dichloracetic  acid, 
13,960  calories  are  liberated.    What  is  the  ratio  of  the  strengths  of  the 
the  two  acids?  Ans.  HN03  :  CHC12COOH  ::  3.1  :  1. 

3.  For  potassium  acetate  we  have  the  following  data:  — 


V 

AV(18°) 

2 

67.1 

10 

78.4 

100 

87.9 

1000 

91.9 

and  fc  =  64.67,  and  L,  =  35.    Compare  the  constants  obtained  by  the 

K'  CHjCOO' 

Ostwald,  Rudolphi  and  Van't  Hoff  dilution  laws. 


394  THEORETICAL  CHEMISTRY 

4.  The  ionization  constant  of  a  0.05  molar  solution  of  acetic  is  0.0000175 
at  18°,  and  0.00001624  at  52°.    Calculate  the  heat  of  ionization  of  the  acid. 
To  what  temperature  does  this  value  correspond? 

Ans.  416  calories  at  35°. 

5.  At  20°  the  specific  conductance  of  a  saturated  solution  of  silver 
bromide  was  1.576  X  10~6  reciprocal  ohms,  and  that  of  the  water  used 
was  1.519  X  10~6  reciprocal  ohms.    Assuming  that  silver  bromide  is 
completely  ionized,  calculate  the  solubility  and  the  solubility  product  of 
silver  bromide,  having  given  that  the  equivalent  conductances  of  potas- 
sium bromide,  potassium  nitrate,  and  silver  nitrate  at  infinite  dilution 
are  137.4,  131.3,  and  121  reciprocal  ohms  respectively. 

Ans.  CAgBr  =  4.49  X  10~7  mol  per  liter,  sAgBr  =  2.03  X  10~13. 

6.  The  solubility  of  silver  cyanate  at  100°  is  0.008  mol  per  liter.     Cal- 
culate the  solubility  in  solution  of  potassium  cyanate  containing  0.1  mol 
of  K'  ions.  Ans.   6.4  X  10~4  mol  per  liter. 

7.  Calculate  the  degree  of  hydrolytic  dissociation  of  a  0.1  molar  solu- 
tion of  ammonium  chloride,  having  given  the  following  data:  —  a,  =  0.86, 
ota  =  0.87,  KNH,OH  =  0.000023,  and  SHSO  =  (0.91  X  10~7)2  at  25°. 

Ans.  x  =  0.006  per  cent. 

8.  In  the  reaction  represented  by  the  equation 

MA3  +  3  H20  =  M  (OH),  +  3  HA, 

the  base  formed  is  insoluble.    Derive  an  expression  for  the  constant  of 
hydrolysis. 

Am.  Kh  =  s-Z£=--^-.^- 
SHA       (I  -  x)   a, 

9.  The  equivalent  conductance  of  aniline  hydrochloride  at  a  dilution 
of  197.6  liters  is  126.7  reciprocal  ohms,  at  25°.    The  equivalent  con- 
ductance of  aniline  hydrochloride  in  the  presence  of  an  excess  of  aniline 
is  106.6;    and  the  equivalent  conductance  of  hydrochloric  acid  at  the 
same  dilution  is  415.    If  the  conductance  of  pure  aniline  is  negligible, 
calculate  the  degree  of  hydrolytic  dissociation  and  the  constant  of  hydrol- 
ysis, assuming  «,  =  aa  =  1. 

Ans.  x  =  6.52  per  cent,  Kh  =  2.33  X  10~5. 

10.  The  hydrolysis  constant  of  aniline  is  2.25  X  10~5,  and  the  ioniza- 
tion constant  is  5.3  X  10~10.    Calculate  the  concentration  of   the  H* 
and  OH'  ions  in  water.  Ans.  CH«  =  CCH'  =  1.09  X  10~7. 


CHAPTER  XVII. 
ELECTROMOTIVE  FORCE. 

Galvanic  Cells.  Since  the  year  1800,  when  Volta  invented 
his  electric  pile,  many  different  forms  of  galvanic  cell  have  been 
introduced. 

It  is  not  our  purpose  to  give  a  detailed  account  of  these  cells, 
but  rather  to  give  a  brief  outline  of  the  theories  which  have  been 
advanced  in  explanation  of  the  electromotive  force  developed  in 
such  cells.  When  two  metallic  electrodes  are  immersed  in  a  solu- 
tion of  an  electrolyte,  a  current  will  flow  through  a  wire  connect- 
ing the  electrodes,  provided  the  two  metals  are  dissimilar,  or  that 
a  difference  exists  between  the  solutions  surrounding  the  electrodes. 
An  electric  current  can  be  obtained  from  a  combination  of  two 
different  metals  in  the  same  electrolyte,  from  two  different  metals 
in  two  different  electrolytes,  from  the  same  metal  in  different  elec- 
trolytes, or  from  the  same  metal  in  two  different  concentrations  of 
the  same  electrolyte. 

In  order  that  the  electromotive  force  of  the  combination  shall 
remain  constant,  it  is  necessary  that  the  chemical  changes  involved 
in  the  production  of  the  current  shall  neither  destroy  the  difference 
between  the  electrodes,  nor  deposit  upon  either  of  them  a  non- 
conducting substance.  A  galvanic  combination  which  fulfils 
these  conditions  very  satisfactorily  is  the  Daniell  cell.  This  cell 
consists  of  zinc  and  copper  electrodes  immersed  in  solutions  of 
their  salts,  as  represented  by  the  scheme 

Zn  -  Sol.  of  ZnS04||  Sol.  of  CuS04  -  Cu, 

in  which  the  two  vertical  lines  indicate  a  porous  partition  separat- 
ing the  two  solutions.  When  the  zinc  and  copper  electrodes  are 
connected  by  a  wire,  a  current  of  positive  electricity  passes  from 
the  copper  to  the  zinc  along  the  wire.  Zinc  dissolves  from  the  zinc 
electrode,  an  equivalent  amount  of  copper  being  displaced  from 

395 


396  THEORETICAL  CHEMISTRY 

the  solution  and  deposited  simultaneously  on  the  copper  electrode. 
As  long  as  only  a  moderate  current  flows  through  the  cell,  the 
original  nature  of  the  electrodes  is  not  modified,  the  only  change 
which  occurs  being  the  gradual  dilution  of  the  copper  sulphate, 
owing  to  the  separation  of  copper  and  its  replacement  by  zinc. 
If  the  loss  of  copper  sulphate  is  replaced,  the  electromotive  force 
of  the  cell  will  remain  constant.  If,  after  the  cell  is  assembled  no 
current  be  allowed  to  flow,  the  copper  sulphate  will  slowly  diffuse 
into  the  solution  of  zinc  sulphate,  and  metallic  copper  will  ulti- 
mately be  deposited  on  the  zinc  electrode.  In  this  way  miniature, 
local  galvanic  cells  will  be  formed  on  the  surface  of  the  zinc,  this 
metal  dissolving  as  though  the  main  circuit  were  closed.  Until  this 
deposition  takes  place,  the  cell  may  be  left  on  open  circuit  with- 
out danger  of  deterioration.  Unless  chemically  pure  zinc  is  used, 
local  action  is  likely  to  occur,  owing  to  the  formation  of  local 
galvanic  couples  between  the  impurities  in  the  electrode,  —  chiefly 
iron,  —  and  the  zinc.  This  action  may  be  prevented  by  amalga- 
mating the  zinc  electrode.  In  this  process  the  mercury  dissolves 
the  zinc  and  not  the  iron,  a  uniform  surface  of  the  former  metal 
being  produced. 

An  interesting  experiment  due  to  Ostwald  *  illustrates  the  con- 
ditions essential  to  the  continuous  production  of  an  electric  current. 
Two  electrodes,  one  of  amalgamated  zinc  and  the  other  of  platinum, 
are  each  immersed  in  a  solution  of  potassium  sulphate,  the  two 
solutions  being  separated  by  a  porous  cup.  When  the  two  elec- 
trodes are  connected  by  means  of  a  wire,  no  permanent  current 
passes.  An  inappreciable  quantity  of  zinc  goes  into  solution, 
since  any  current  must  necessarily  first  liberate  potassium  at  the 
platinum  electrode,  the  potassium  thus  set  free  reacting  with  the 
water.  This  process  requires  the  expenditure  of  more  energy 
than  the  solution  of  the  zinc  supplies.  If  sulphuric  acid  is  added 
to  the  compartment  containing  the  zinc,  the  condition  of  the 
system  will  be  unchanged,  the  zinc  remaining  undissolved.  If, 
on  the  other  hand,  a  few  drops  of  sulphuric  acid  are  added  to  the 
compartment  containing  the  platinum  electrode,  bubbles  of 
hydrogen  will  appear  and  the  zinc  will  dissolve  with  the  simulta- 
*  Phil.  Mag.  [5],  32,  145  (1891). 


ELECTROMOTIVE  FORCE  397 

neous  development  of  an  electric  current.  This  experiment  shows 
that  in  order  that  positively  charged  ions  may  enter  a  solution, 
an  equivalent  amount  of  negatively  charged  ions  must  be  intro- 
duced, or  an  equivalent  amount  of  positively  charged  ions  must 
be  removed. 

Reversible  Cells.  Galvanic  cells  are  either  reversible  or  non- 
reversible,  according  as  the  processes  taking  place  within  them  can 
be  reversed  or  not.  If  we  disregard  the  slow  processes  of  diffusion, 
the  Daniell  cell  may  be  taken  as  an  example  of  an  almost  perfect 
reversible  element.  If  an  electromotive  force  slightly  less  than 
that  of  the  cell  be  applied  to  it  in  the  reverse  direction,  the  current 
within  the  cell  will  flow  from  the  zinc  to  the  copper  electrode  as 
usual.  On  the  other  hand,  if  the  external  electromotive  force 
slightly  exceeds  that  of  the  cell,  the  current  within  the  cell  will 
flow  in  the  reverse  direction,  zinc  being  deposited  and  copper 
dissolved. 

Any  cell  from  which  gas  is  evolved  is  non-reversible,  since  the 
passage  of  a  current  in  the  reverse  direction  cannot  restore  the 
cell  to  its  original  condition. 

Relation  between  Chemical  Energy  and  Electrical  Energy. 
Helmholtz  and  Thomson  were  the  first  to  propose  a  theory  of  the 
action  of  the  reversible  cell.  According  to  this  theory  the  energy 
of  the  chemical  process  taking  place  within  the  cell  is  considered 
as  completely  transformed  into  electrical  energy.  It  was  soon 
shown  that  this  theory  is  inadequate,  since  with  the  exception  of 
the  Daniell  cell,  the  chemical  energy  is  not  equivalent  to  the  elec- 
trical energy  produced.  Subsequently,  Gibbs  *  and  Helmholtz  f 
showed  independently  that  only  in  those  cells  in  which  the  elec- 
tromotive force  does  not  vary  with  the  temperature,  is  the  chem- 
ical energy  completely  transformed  into  electrical  energy.  They 
also  derived  an  equation  expressing  the  relation  between  the 
chemical  and  electrical  energies  in  any  reversible  cell.  Let  us 
imagine  a  reversible  element  in  which  an  amount  of  heat  q,  is  either 
liberated  or  absorbed,  when  one  faraday  of  electricity  has  passed 
through  the  cell.  Let  the  cell  be  immersed  in  a  bath,  which  is  so 

*  Proc.  Conn.  Acad.,  3,  501  (1878). 

t  Sitzungsbericht.,  Ber.  Akad.,  22,  825  (1882). 


398  THEORETICAL  CHEMISTRY 

arranged  that  the  temperature  of  the  cell  can  be  maintained  con- 
stant under  any  working  conditions.  If  the  chemical  process 
within  the  cell  is  accompanied  by  an  evolution  or  an  absorption 
of  heat,  then  of  necessity,  heat  energy  must  be  removed  or 
supplied  in  order  to  maintain  the  temperature  of  the  system 
constant.  It  is  evident  that  this  will  involve  a  corresponding 
decrease  or  increase  in  the  electrical  energy  produced  by  the  cell. 
The  effect  of  the  evolution  or  absorption  of  heat  upon  the 
electrical  energy  of  the  cell  may  be  derived  in  the  following 
manner:  Let  the  cell  be  heated  from  its  initial  temperature  T 
to  the  temperature  (T  +  dT),  and  let  the  corresponding  change 
in  the  electromotive  force  of  the  cell  be  dir.  If  now  the  circuit  be 
closed  and  one  faraday  of  electricity  be  allowed  to  pass  through 
the  cell,  F  (TT  +  dir)  units  of  electrical  work  will  be  done.  In 
order  that  the  temperature  of  the  cell  may  not  change,  (q  +  dq) 
units  of  heat  must  be  absorbed.  The  cell  is  now  cooled  to  the 
temperature  T,  at  which  the  electromotive  force  of  the  cell  is  TT, 
and  F  units  of  electricity  are  sent  through  the  cell  in  the  reverse 
direction,  thus  increasing  the  energy  of  the  cell  by  Fir.  In  order 
to  maintain  the  temperature  of  the  cell  unchanged,  q  units  of 
heat  must  be  removed.  If  the  cell  is  completely  reversible,  when 
this  cycle  of  operations  is  completed,  it  will  be  restored  to  its 
original  condition.  The  total  work  done  during  the  cycle  is 
F  (TT  +  dir)  —  FTT,  and  the  amount  of  heat  transformed  into  work 
is  (q  -+-  dq)  —  q]  therefore,  applying  the  second  law  of  thermo- 
dynamics, we  have 

dq  =  Fdir      dT 
q~~      q       "   T' 

or 

*  =  FT%.  (1) 

Since  the  electrical  energy  is  equal  to  FTT,  the  relation  between 
this  and  Q,  the  chemical  energy  of  the  cell,  expressed  in  calories, 
becomes 

F*  =  Q  +  q.  (2) 


ELECTROMOTIVE  FORCE  399 

Substituting  in  equation  (2)  the  value  of  q  given  in  equation  (1), 
we  obtain 


or 

=  6+T^l.  m 

F         d,T* 

When  -T~  =  0,  TT  becomes  equal  to  ^ ,  or,  when  the  temperature 

coefficient  of  the  cell  is  zero,  the  electrical  energy  is  equal  to  the 
chemical  energy.  This  is  true  of  the  Daniell  cell,  which  has  an 
extremely  small  temperature  coefficient. 

For  cells  in  which  the  electromotive  force  varies  appreciably 
with  the  temperature,  it  is  possible  to  calculate  the  value  of  the 
electromotive  force  at  any  temperature  by  means  of  the  Gibbs- 
Helmholtz  equation,  provided  the  temperature  coefficient  is  known. 
In  the  Grove  gas  cell,  TT  =  1.062  and  Q  =  34,200  calories,  hence 

34'2°°  -0.418- 


d  96,540  X  0.2394 

The  value  determined  by  direct  experiment  is  —  0.416  volt.  The 
Gibbs-Helmholtz  equation  shows  that  the  amount  of  heat  accom- 
panying a  chemical  process  does  not  alone  furnish  a  measure  of 
the  electrical  energy  which  may  be  obtained  from  it,  since  the 
heat  which  is  absorbed  from  the  surrounding  medium  may  also 
be  transformed  into  electrical  energy,  or  the  output  of  electrical 
energy  may  be  less  than  the  heat  evolved  by  the  chemical 
reaction  within  the  cell. 

Solution  Pressure.  It  is  a  familiar  fact  that  water  has  a 
tendency  to  assume  the  form  of  vapor,  and  if  the  vapor  be  contin- 
ually removed  from  its  surface,  a  definite  mass  of  water  will  grad- 
ually be  completely  transformed  into  the  state  of  vapor.  The 
pressure  of  the  vapor  at  any  one  temperature  is  a  measure  of  the 
tendency  of  water  to  undergo  this  transformation.  This  tendency 
of  water  to  assume  another  form  than  that  in  which  it  actually 
exists,  is  typical  of  all  substances.  Attention  has  already  been 
directed  to  this  fact  in  connection  with  the  application  of  the  law 


400  THEORETICAL  CHEMISTRY 

of  mass  action  to  heterogeneous  equilibria.  It  was  then  pointed 
out  that  all  solids  have  a  definite  vapor  pressure  at  a  definite 
temperature,  which  is  independent  of  the  amount  of  solid  present. 
When  a  solid,  such  as  cane  sugar,  is  brought  in  contact  with  water, 
it  tends  to  pass  into  solution.  This  tendency  is  constant  at 
constant  temperature,  since  the  active  mass  of  the  solid  is  constant. 
From  the  close  analogy  between  the  vapor  state  and  the  dissolved 
state,  the  tendency  of  a  solid  to  pass  into  solution  is  termed  the 
solution  pressure.  A  dissolved  solid,  on  the  other  hand,  also  shows 
a  tendency  to  separate  from  the  solution  as  the  concentration 
is  increased.  When  the  solution  becomes  supersaturated,  the 
tendency  of  the  solute  to  separate  in  the  solid  form  is  greater  than 
the  tendency  of  the  solid  to  dissolve.  It  is  evident  from  these 
considerations  that  the  pressure  exerted  by  the  dissolved  solid 
is  its  osmotic  pressure,  and  whether  the  solid  will  dissolve  or 
separate  from  the  solution  depends  upon  whether  the  solution 
pressure  is  greater  or  less  than  the  osmotic  pressure. 

This  conception  of  solution  pressure  was  introduced  by  Nernst,* 
and  in  conjunction  with  the  theory  of  electrolytic  dissociation  it 
has  proved  of  great  value  in  affording  a  much  deeper  insight  into 
the  mechanism  of  the  development  of  differences  in  potential 
within  a  galvanic  cell.  Thus,  when  a  metal  is  dipped  into  water 
it  tends  to  dissolve  owing  to  its  solution  pressure  P  and,  in  con- 
sequence of  this  tendency,  it  sends  a  certain  number  of  positive 
ions  into  solution.  The  solution  thus  becomes  positively  charged, 
and  the  metal,  which  was  initially  neutral,  acquires  a  negative 
charge  due  to  the  loss  of  a  certain  amount  of  positive  electricity. 
This  process  will  cease  when  the  solution  becomes  so  strongly 
charged  with  positive  electricity  that  it  prevents  the  separation 
of  any  more  positive  ions  from  the  metal.  Relatively  few  ions 
leave  the  metal  before  equilibrium  is  established,  since  the  charge 
on  each  ion  is  so  great;  in  fact,  the  concentration  of  metal  ions  in 
the  solution  is  much  too  small  to  be  detected  analytically.  When 
a  metal  is  dipped  into  a  solution  of  one  of  its  salts,  the  conditions 
are  altered.  In  this  case,  the  positive  ions  of  the  metal  already 
present  in  the  solution  oppose  the  entrance  of  more  positive  ions, 
*  Zeit.  phys.  Chem.,  4,  150  (1889). 


ELECTROMOTIVE   FORCE 


401 


and  the  equilibrium  between  these  two  opposing  tendencies  will 
be  conditioned  by  the  relative  values  of  the  solution  pressure  P, 
of  the  metal,  and  the  osmotic  pressure  p,  of  the  ions  of  the  dissolved 
salt. 

It  is  evident  that  the  three  following  conditions  are  possible:  — 

(1)  If  P  >  p,  the  metal  will  continue  to  send  ions  into  the 
solution  until  the  accumulated  charges  in  the  solution  oppose 
further  action.     The  solution  acquires  a  positive  charge  and  the 
metal  a  negative  charge. 

(2)  If  P  <  p,  the  positive  ions  of  the  dissolved  salt  will  sepa- 
rate on  the  metal  until  the  accumulated  charges  oppose  further 
action.     The  metal  acquires  a  positive  charge  and  the  solution  a 
negative  charge. 

(3)  If  P  =  p,  no  action  will  take  place  and  no  difference  of 
potential  will  be  established  between  the  metal  and  the  solution. 
These  three  cases  are  represented  diagrammatically  in  Fig.  90. 


Fig.  90. 

When  equilibrium  is  established  and  the  metal  is  negative  against 
the  solution,  the  metal  is  surrounded  by  a  layer  of  positively 
charged  ions.  This  constitutes  what  is  known  as  a  Helmholtz 
electrical  double  layer.  If  positive  electricity  be  communicated  to 
the  metal,  the  double  layer  will  be  broken  and  more  ions  will 
pass  from  the  metal  into  the  solution,  but  as  soon  as  the  supply 


402 


THEORETICAL  CHEMISTRY 


of  positive  electricity  is  cut  off,  the  double  layer  will  again  be 
formed.  Similarly,  when  the  metal  is  positive  against  the  solu- 
tion, an  electrical  double  layer  will  be  formed,  the  metal  being 
surrounded  by  a  layer  of  negatively  charged  ions. 

The  actual  existence  of  a  Helmholtz  double  layer  has  been 
demonstrated  by  Palmaer.*  In  his  experiments,  Palmaer  allowed 
exceedingly  minute  globules  of  mercury  to  fall  into  a  dilute  solu- 
tion of  mercurous  nitrate  contained  in  a  tall  vessel,  the  bottom 
of  which  was  covered  with  a  layer  of  pure  mercury,  as  shown  in 
Fig.  91.  Since  the  solution  pressure  of  mercury  is  less  than  the 


Fig.  91. 

osmotic  pressure  of  the  Hg*  ions,  each  drop  of  mercury  as  it 
enters  the  solution  will  acquire  a  positive  charge,  and  if  the  theory 
of  the  electrical  double  layer  is  correct,  this  positively  charged 
globule  should  attract  negatively  charged  ions  and  drag  them  down 
through  the  solution.  When  the  globule  reaches  the  mercury  at 
*  Zeit.  phys,  Chem.,  25,  265  (1898);  28,  257  (1899);  36,  664  (1901). 


ELECTROMOTIVE  FORCE  403 

the  bottom  of  the  vessel,  it  will  give  up  its  positive  charge  and  as 
many  Hg"  ions  will  pass  into  solution  as  there  are  N(V  ions  in  the 
double  layer.  The  solution  will  thus  become  more  concentrated 
just  above  the  layer  of  mercury  on  the  bottom  of  the  vessel.  Pal- 
maer's  experiments  showed  that  this  difference  in  concentration 
is  actually  produced,  in  some  cases  the  concentration  in  the  upper 
part  of  the  solution  being  reduced  as  much  as  50  per  cent. 

The  metals  sodium,  potassium,  .  .  .  zinc,  cadmium,  cobalt, 
nickel,  and  iron  are  negative  against  solutions  of  their  salts,  or 
P  >  p.  The  noble  metals  are  generally  positive  against  solutions 
of  their  salts,  or  P  <  p.  The  anions  are,  so  far  as  is  known,  posi- 
tive to  solutions  of  their  salts.  Electrolytic  solution  pressure 
varies  with  the  temperature,  with  the  nature  of  the  solvent,  and 
also  with  the  concentration  of  the  active  substance  in  the  elec- 
trode. 

The  Difference  of  Potential  between  a  Metal  and  a  Solution. 
From  the  foregoing  considerations,  it  is  possible  to  derive  an 
equation  expressing  the  difference  of  potential  between  a  metallic 
electrode  and  a  solution  of  one  of  its  salts. 

Let  us  imagine  one  ion  of  a  metal  to  be  transferred  from  the 
electrolytic  solution  pressure  P,  to  the  osmotic  pressure  p.  The 
osmotic  work  done  will  be 


Integrating  this  expression,  we  have 

p 
Osmotic  work  =  RT  log«  —  • 

The  corresponding  electrical  energy  gained  is  nFv,  where  TT  is  the 
difference  of  potential  between  the  metal  and  the  solution,  F  =  1 
faraday  =  96,540  coulombs,  and  n  is  the  valence  of  the  metal. 
Since  the  osmotic  work  done  is  equivalent  to  the  electrical  energy 
gained,  we  may  equate  these  two  expressions,  as  follows:  — 

nFir  =  RT  \oge-> 
p 

or 

RT 


404  THEORETICAL  CHEMISTRY 

Expressing  both  sides  of  equation  (1)  in  electrical  units,  and  trans- 
forming to  Briggsian  logarithms,  we  obtain 


96,540  X  n  X  0.4343  X  0.2394        &  p 
or 


For  univalent  ions  at  17°,  we  have 

TT  =  0.0575  log--  (3) 

In  a  galvanic  cell  composed  of  two  metals,  each  immersed  in  a 
solution  of  one  of  its  salts,  a  difference  of  potential  may  be  estab- 
lished (1)  at  the  junction  of  the  two  metals,  (2)  at  the  junction  of 
the  two  solutions,  and  (3)  at  the  points  of  contact  of  the  metals 
with  their  respective  'solutions.  If  the  temperature  remains  con- 
stant, (1)  is  negligible,  and  in  general,  (2)  is  exceedingly  small; 
therefore,  the  electromotive  force  of  the  cell  may  be  considered  as 
due  to  the  differences  of  potential  arising  at  the  two  electrodes. 
Assuming  the  temperature  to  be  17°,  the  electromotive  force  of 
the  cell  will  be 

0.0575,      Pi      0.0575,      P2 

7T  =  TTi  —  7T2  =  log log • 

n         *pi  n        *p2 

The  Measurement  of  Electromotive  Force.  The  value  of  the 
electromotive  force  of  a  cell  may  vary  with  the  conditions  of  meas- 
urement. Since,  according  to  Ohm's  law,  E  =  C  (R  +  r),  where 
R  is  the  resistance  of  the  external  circuit  and  r  is  the  internal 
resistance  of  the  cell,  it  follows  that  the  fall  of  potential  CR,  in 
the  external  circuit,  will  only  be  equal  to  E  when  r  is  negligible 
in  comparison  with  R.  Furthermore,  when  the  circuit  is  closed, 
the  electrodes  of  the  cell  frequently  become  polarized,  owing  to 
the  deposition  of  the  products  of  electrolysis,  and  an  opposing 
electromotive  force  is  set  up. 

To  avoid  these  difficulties,  the  electromotive  force  is  usually 
measured  on  open  circuit  by  the  Poggendorff  compensation  method. 
In  this  method  the  electromotive  force  to  be  measured  is  just 


ELECTROMOTIVE  FORCE 


405 


balanced  by  an  equal  and  opposite  electromotive  force,  so  that  no 
current  passes.  The  arrangement  of  the  apparatus  for  such 
measurements  is  shown  in  Fig.  92.  If  the  two  ends  of  the  wire  A B 


Fig.  92. 

of  a  Wheatstone  bridge  are  connected  to  a  lead  accumulator  C, 
there  will  be  a  uniform  fall  of  potential  along  its  length.  The 
amount  of  fall  along  any  portion  AD  will  be  proportional  to  the 

AD 
length  AD,  and  equal  to  the  fraction-^  of  the  total  fall  of  poten- 


tial  along  the  entire  length  of  the  wire.  Now  let  one  terminal  of 
a  cell  whose  electromotive  force  is  less  than  that  of  C  be  con- 
nected to  A,  and  the  other  terminal  be  connected  through  a 
galvanometer  G,  with  a  sliding  contact  D,  the  two  cells  E  and 
C  working  in  opposition.  A  current  will  flow  through  the  cir- 
cuit AEGD,  and  will  be  indicated  by  the  galvanometer  at  all 
positions,  except  that  at  which  the  fall  of  potential  along  the  wire 
from  A  to  D  is  equal  to  the  electromotive  force  of  E.  Hence  we 

have 

e.m.f.  of  C  :  e.m.f.  of  E  ::  AB  :  AD, 

from  which  the  value  of  the  electromotive  force  of  the  cell  E,  can 
be  calculated.  Since  the  electromotive  force  of  a  lead  accumu- 
lator is  not  quite  constant,  it  is  customary,  after  having  deter- 
mined the  point  D,  to  substitute  a  standard  cell  for  E,  and  balance 


406 


THEORETICAL  CHEMISTRY 


this  against  the  accumulator,  finding  a  new  point  of  balance  ZX. 
We  now  have  the  proportion 

e.m.f.  of  C  :  e.m.f.  of  standard  ::  AB  :  AD'. 
Combining  these  two  proportions,  we  obtain 

e.m.f.  of  E  :  e.m.f.  of  standard  ::  AD  :  AD'. 

Instead  of  using  a  galvanometer  as  a  "null"  instrument  for  indi- 
cating when  the  point  of  balance  has  been  reached,  it  is  prefer- 
able to  make  use  of  a  capillary  electrometer. 

Standard  Cells.  It  is  apparent  that  the  accuracy  of  all 
measurements  of  electromotive  force  is  dependent  upon  the  cell 
employed  as  a  standard.  Much  time  has  been  devoted  to  the 
study  of  various  reversible  elements  with  a  view  to  establishing  a 
standard  of  electromotive  force.  As  a  result  we  have  the  com- 
plete specifications  for  two  standard  cells,  either  of  which  may  be 
readily  reproduced. 

(a)  The  Weston,  or  Cadmium  Standard  Cell.  The  most  widely 
used  standard  of  electromotive  force  is  the  so-called  Weston  cell, 
made  up  according  to  the  scheme 

Hg  -  Solution  Hg2S04  [|  Solution  CdS04  -  Cd. 
A  diagram  of  the  usual  form  of  the  Weston  cell  is  given  in  Fig.  93. 


Fig.  93. 


ELECTROMOTIVE  FORCE  407 

A  short  platinum  wire  is  sealed  through  the  bottom  of  each  limb 
of  the  H-shaped  vessel.  In  one  limb  is  placed  a  small  amount  of 
a  10  to  15  per  cent  cadmium  amalgam,  A;  B  is  a  layer  of  small 

o 

crystals  of  CdS04  •  ^  H2O.     In  the  other  limb  is  placed  a  small 

o 

amount  of  pure  mercury,  over  which  is  a  layer,  D,  of  a  paste 
composed  of  solid  mercurous  sulphate  and  a  saturated  solution 
of  cadmium  sulphate.  The  cell  is  then  filled  with  crystals  of 
cadmium  sulphate  and  a  saturated  solution  of  cadmium  sulphate. 
The  two  limbs  of  the  cell  are  closed  with  a  thin  layer  of  paraffin 
E,  cork  Fj  and  sealing  wax  G.  If  carefully  prepared,  this  cell  will 
remain  unaltered  for  years  and  will  have  an  electromotive  force 
at  20°  of  1.0183  volts.  In  addition  to  the  fact  that  it  can  be  so 
easily  reproduced,  the  temperature  coefficient  of  the  cell  is  almost 
negligible. 

The  electromotive  force  of  a  Weston  standard  cell  at  any  temper- 
ature t,  is  given  by  the  formula 

e.m.f.  at  t°  =  1.0183  -  0.000038  (t  -  20). 

(b)  The  Clark,  or  Zinc  Standard  Cell.  Until  about  ten  years 
ago,  the  Clark  cell  was  considered  to  be  the  most  trustworthy 
standard  of  electromotive  force.  This  cell  is  made  up  according 
to  the  scheme 

Hg  -  Solution  Hg2S04 1|  Solution  ZnS04  -  Zn. 

The  construction  of  the  cell  is  similar  to  that  of  the  Weston  cell. 
It  may  be  reproduced  with  great  accuracy  and  with  no  more 
trouble  than  the  Weston  cell,  but  its  relatively  large  temperature 
coefficient  renders  it  less  satisfactory.  The  electromotive  force 
of  the  Clark  standard  cell  at  any  temperature  t,  may  be  calculated 
by  means  of  the  formula 

e.m.f.  at  t°  =  1.4328  -  0.00119  (t  -  15)  -  0.00007  (t  -  15)2. 

The  Capillary  Electrometer.  When  pure  mercury  is  covered 
with  sulphuric  acid,  its  surface  tension  is  diminished.  This  may 
be  shown  by  the  following  experiment:  In  a  small  evaporating 
dish  place  about  5  cc.  of  pure  mercury,  and  cover  it  with  a  10  per 


408 


THEORETICAL  CHEMISTRY 


cent  solution  of  sulphuric  acid  to  which  has  been  added  enough 
potassium  dichromate  to  impart  a  light  yellow  color  to  the  so- 
lution. The  globule  of  mercury  will  immediately  flatten  out, 
indicating  that  its  surface  tension  has  diminished.  If  now  the  mer- 
cury be  touched  with  a  piece  of  iron  wire,  it  will  instantly  contract 
until  the  contact  with  the  wire  is  broken;  it  will  then  flatten  out, 
until  it  again  comes  in  contact  with  the  wire,  when  the  globule 
of  mercury  will  once  more  contract.  In  this  way  a  regular  pul- 
sation of  the  mercury  may  be  obtained.  This  interesting  phenom- 
enon was  observed  early  in  the  nineteenth  century  by  Henry,  but 
was  first  satisfactorily  explained  by  Lippmann  *  in  1873.  Lipp- 
mann  showed  that  when  the  globule  of  mercury  is  negatively 
electrified,  its  surface  tension  increases  and  the  drop  shrinks. 
If  sufficient  negative  electricity  is  imparted  to  the  mercury  it 
is  possible  to  restore  the  globule  to  its  original  form.  On  ap- 
plying more  negative  electricity,  the  globule  of  mercury  again 
expands.  When  the  iron  wire  touches  the  globule  it  charges 
it  negatively,  because  when  the  iron  dis- 
solves, it  furnishes  positively  charged  ions 
to  the  solution  and  thus  acquires  a  negative 
charge  which  it  imparts  to  the  mercury.  At 
B I -LI  the  same  time,  the  chromic  acid  in  the  so- 

lution undergoes  reduction  to  chromium 
sulphate.  Lippmann  concluded  from  his 
experiments  that  the  difference  of  poten- 
tial arises  at  the  surface  of  contact  between 
the  mercury  and  the  solution  of  the  electro- 
/• — j  lyte,  and  that  the  surface  tension  of  the 

C  mercury  is  a  function  of  the  difference  of 

J     L         potential.     Making  use  of  this  principle  he 
f         \       constructed  the  capillary  electrometer,  a  con- 
^••^A     venient  form  of  which  is  shown  in  Fig.  94. 
The  bulb  A,  through  the  bottom  of  which 
is  sealed   a   platinum  wire,  contains  pure 
mercury  and  dilute  sulphuric  acid  (1  :  6). 
Pure  mercury  is  poured  into  the  other  limb  of  the  electrometer 
*  Pogg.  Ann.,  149,  546  (1873). 


Fig.  94. 


ELECTROMOTIVE  FORCE 


409 


until  it  stands  at  B  in  that  tube,  and  at  C  in  the  capillary  tube. 
Owing  to  the  capillary  depression  of  the  mercury,  C  lies  below  B* 
Electrical  connection  with  the  mercury  at  B  is  established  by 
means  of  a  platinum  wire. 

The  position  of  the  mercury  in  the  capillary  is  determined  by 
its  surface  tension;  if  the  surface  tension  is  increased,  the  mercury 
will  descend;  if  it  is  diminished,  the  mercury  will  ascend.  If  a, 
negative  charge  is  communicated  to  the  mercury  at  B,  the  surface 
tension  will  be  increased  and  the  meniscus  will  descend ;  if  a  posi- 
tive charge  is  imparted  to  the  mercury  at  B}  the  surf  ace  tension 
will  be  diminished  and  the  meniscus  will  ascend. 

The  amplitude  of  the  movement  of  the  meniscus  is  an  inverse 
function  of  the  diameter  of  the  capillary  tube.     If  the  meniscus 
be  observed   through   a   microscope 
provided  with  an  eye-piece  microm- 
eter, Fig.  95,  it  is  possible  to  detect 
very  slight  movements,  and  to  meas- 
ure differences  of  potential  less  than 
0.0001  volt. 

The  capillary  electrometer  is  an  ex- 
cellent "null"  instrument.  In  using 
the  electrometer  no  large  electromo- 
tive force  should  be  applied,  since 
the  meniscus  surface  becomes  polar- 
ized very  easily.  If  this  should  occur, 
a  new  surface  may  be  secured  by 
blowing  gently  at  B  and  forcing  a 
drop  of  mercury  out  of  the  capillary 
into  the  bulb.  Lippmann  studied  the 
effect  of  steadily  increasing  potentials 
on  the  movement  of  the  meniscus. 
Plotting  movements  of  the  meniscus 
on  the  axis  of  ordinates,  and  potentials 

on  the  axis  of  abscissae,  he  found  that  there  is  a  maximum  in  the 
curve  corresponding  to  about  0.8  volt.  This  is  the  electromotive 
force  which  must  be  applied  in  order  to  counterbalance  the  differ- 
ence of  potential  produced  by  the  contact  of  dilute  sulphuric  acid 


Fig.  95. 


410 


THEORETICAL  CHEMISTRY 


Sulphuric  Acid- 


Mercury- 


Fig.  96. 


with  the  surface  of  the  mercury.     At  the  meniscus  surface  an 
electrical  double  layer  is  formed.     The  mercury   is  positively 

charged,  and  above  it  there  must  be  a 
layer  of  negatively  charged  ions,  as 
shown  in  Fig.  96.  Just  how  this  double 
layer  is  formed  is  not  known  with  cer- 
tainty, but  it  has  been  suggested  that 
the  slight  film  of  oxide  which  is  prob- 
ably present  on  the  surface  of  the  pur- 
est mercury,  dissolves  in  the  sulphuric 
acid  forming  a  solution  of  mercurous 
sulphate,  and  from  this  solution  the 
positively  charged  Hg*  ions  deposit  on 
the  mercury,  giving  it  a  positive  charge. 
Whether  this  explanation  is  correct  or 
not,  the  fact  remains  that  the  mercury 
is  positive  against  the  solution. 

Normal  Electrodes.  The  method  commonly  employed  for 
the  measurement  of  the  difference  of  potential  between  a  metal 
and  a  solution,  is  based  upon  the  use  of  an  electrode  in  which  the 
difference  of  potential  between  the  electrode  and  a  certain  solution 
of  one  of  its  salts  is  known.  Such  an  electrode  is  called  a  normal 
electrode.  If  a  cell  is  made  up  by  combining  the  normal  electrode 
with  the  electrode  whose  potential  is  to  be  determined,  it  is  possible, 
from  measurements  of  the  resulting  electromotive  force,  to  cal- 
culate the  value  of  the  unknown  difference  of  potential.  The  most 
convenient  electrode  to  prepare  is  the  normal  calomel  electrode,  a 
satisfactory  form  of  which  is  shown  in  Fig.  97.  The  bottom  of 
the  electrode  vessel  is  covered  with  a  layer  of  pure  mercury,  upon 
which  is  poured  a  paste,  prepared  by  rubbing  together  in  a  mortar 
mercury  and  calomel,  moistened  with  a  molar  solution  of  potassium 
chloride.  The  vessel  is  then  filled  with  a  molar  solution  of  the 
same  salt  which  has  been  saturated  with  calomel  by  prolonged 
shaking  with  the  latter.  Connection  with  the  mercury  is  estab- 
lished by  means  of  a  platinum  wire  sealed  into  a  glass  tube  A ,  the 
latter  being  passed  through  the  rubber  stopper  which  closes  the 
vessel.  In  using  the  calomel  electrode,  the  bent  side  tube  C  is 


ELECTROMOTIVE  FORCE 


411 


filled  with  molar  potassium  chloride  by  applying  suction  at  the 
side  tube  B,  which  is  then  closed  by  means  of  a  pinch-cock. 

The  difference  of  potential,  at  any  temperature  t,  of  the  calo- 
mel electrode  prepared  as  described,  and  represented  by  the 
scheme 

Hg  -  Solution  HgCl  in  molar  KC1, 
is 

TT  =  -0.560  5 1  +  0.0006  (t  -  18) }  volt. 

The  negative  sign  indicates  that  the  solution  is  negative  to  the 
electrode.    In  order  to  measure  the  potential  of  another  electrode 


W 


Fig.  97. 


Fig.  98. 


by  means  of  the  calomel  electrode,  the  arrangement  shown  in 
Fig.  98  is  commonly  used.  Here  A  represents  the  "half -element" 
of  which  the  potential  is  to  be  determined,  B  represents  the  side 
tube  of  the  calomel  electrode,  and  C  represents  an  intermediate, 
connecting  vessel  containing  a  molar  solution  of  potassium  chloride. 
In  cases  where  potassium  chloride  forms  a  precipitate  with  the 
electrolyte  in  A,  the  solution  in  C  may  be  replaced  by  a  molar 


412  THEORETICAL  CHEMISTRY 

solution  of  potassium  nitrate  without  altering  the  value  of  the 
electromotive  force  of  the  cell.  The  original  measurement  of 
the  potential  of  the  calomel  electrode  was  made  by  forming  a 
cell  with  this  and  another  electrode  whose  potential  against  its 
solution  is  zero.  Such  an  electrode  is  known  as  a  null  electrode. 
Thus,  if  a  copper  electrode  is  immersed  in  a  solution  of  copper 
sulphate,  the  Cu"  ions  will  leave  the  solution  and  charge  the 
electrode  positively.  If  now  a  solution  of  potassium  cyanide  is 
added,  the  nearly  undissociated  salt,  K2Cu2  (CN)4,  will  be  formed, 
and  by  adding  a  sufficient  amount  of  the  solution,  the  concentration 
of  the  Cu"  ions  may  be  reduced  until  the  metal  and  the  solution 
have  the  same  potential.  The  addition  of  more  potassium  cyanide 
will  still  further  diminish  the  osmotic  pressure  of  the  Cu"  ions, 
and  the  electrode  will  acquire  a  negative  charge.  Similarly,  mer- 
cury in  a  solution  of  a  double  cyanide  may  be  used  as  a  null  elec- 
trode. 

Another  form  of  null  electrode  is  the  so-called  dropping  elec- 
trode of  Helmholtz.*  The  principle  involved  in  this  electrode 
has  already  been  discussed  in  connection  with  Palmaer's  experi- 
ment (p.  402).  An  extremely  fine  stream  of  mercury  is  allowed 
to  flow  from  a  funnel  having  a  minute  capillary  orifice:  the  stem 
of  the  funnel  dips  below  the  surface  of  a  molar  solution  of  potas- 
sium chloride  containing  mercurous  ions.  As  each  little  globule 
enters  the  solution,  it  acquires  a  positive  charge  and  attracts  the 
negatively  charged  ions  of  the  electrolyte,  dragging  them  down 
with  itself.  When  the  globule  reaches  the  layer  of  mercury  at 
the  ^bottom  of  the  vessel,  its  surface  and  capacity  are  diminished, 
and  as  many  Hg*  ions  leave  the  layer  of  mercury  and  enter  the 
solution  as  there  were  negatively  charged  ions  carried  down  by 
the  globule.  This  process  continues  until  the  osmotic  pressure 
of  the  remaining  ions  is  equal  to  the  solution  pressure  of  the  metal: 
the  mercury,  both  in  the  stream  and  at  the  bottom  of  the  vessel, 
has  the  same  potential  as  the  solution.  If  now  the  difference  of 
potential  between  the  mercury  in  the  funnel  and  the  mercury  in 
the  vessel  be  measured,  we  shall  obtain  the  potential  of  mercury 
against  a  molar  solution  of  potassium  chloride.  The  dropping 
*  Ann.  der  Phys.,  44,  42  (1890). 


ELECTROMOTIVE  FORCE 


413 


electrode  was  for  a  long  time  regarded  as  an  ideal  standard  of 
potential,  but  Nernst  has  quite  recently  pointed  out  a  number 
of  serious  objections  to  it.  Until  a  wholly  satisfactory  standard 
of  potential  is  obtained,  he  proposes  that  the  potential  of  the 
hydrogen  electrode  be  adopted  as  the  standard.  This  consists 
of  a  strip  of  platinized  platinum,  half  in  pure  hydrogen  gas  and 
half  in  a  solution  of  sulphuric  acid  of  such  concentration  that 
it  shall  contain  1  gram  of  hydrogen  ions  per  liter.  The  use  of 
the  hydrogen  electrode  as  a  standard  is,  of  course,  purely  arbi- 
trary, but  there  are  many  advantages  in  referring  differences  of 
potential  to  this  standard.  Owing  to  certain  experimental  diffi- 
culties attending  the  use  of  this  electrode,  it  is  customary  to 
make  the  actual  measurements  with  the  calomel  electrode,  and 
then  refer  them  to  the  hydrogen  standard,  taking  the  potential 
of  the  calomel  electrode  to  be  —  0.283  volt  when  referred  to 
the  hydrogen  electrode  as  zero.  The  negative  sign  indicates  that 
the  solution  is  negative  against  the  electrode.  The  following  table 
contains  some  of  the  values  obtained  by  Wilsmore  *  for  differ- 
ent metals  immersed  in  solutions  of  their  salts,  of  such  concentra- 
tion that  they  contain  an  equivalent  weight  in  grams  of  the 
ions  of  the  metal  per  liter. 


Element. 

Calomel  Electrode 
=  -0.283  volt. 
Hydrogen  Elec- 
trode, 0.0. 

Calomel  Electrode 
=  -0.56  volt. 
Absolute 
Potentials. 

Manganese                                        

1.075 

0.798 

Zinc 

0.770 

0.493 

Cadmium  

0.420 

0.143 

Iron  

0.344 

0.067 

Thallium  

0.322 

0.045 

Cobalt  

0.232 

-0.045 

Nickel    

0.228 

-0.049 

Lead  ,  

0.151 

-0.132 

Hydrogen                                                     .    . 

0.0 

-0.277 

Copper 

-0.329 

-0.606 

Mercury 

-0  753 

-1.030 

Silver 

—0  771 

-1.048 

Chlorine  

-1.353 

-1.636 

Bromine     

-0.993 

-1.270 

Iodine  

-0.520 

-0.797 

Zeit.  phys.  Chem.,  35,  291  (1900). 


414  THEORETICAL  CHEMISTRY 

It  is  apparent  that  the  potential  difference  between  a  metal 
and  a  solution  of  one  of  its  salts  is  independent  of  the  anion. 
This  table  enables  us  to  compute  the  electromotive  force  of  any 
cell  we  might  construct.  Thus,  suppose  a  cell  to  be  made  up 
consisting  of  a  zinc  electrode  immersed  in  a  solution  containing  an 
equivalent  weight  in  grams  of  zinc  ions,  and  a  copper  electrode 
immersed  in  a  solution  containing  an  equivalent  weight  in  grams 
of  copper  ions.  According  to  the  data  given  in  the  table,  the 
electromotive  force  of  the  cell  will  be 

TT  =  0.329  -  (-  0.770)  =  1.099  volts. 

The  electromotive  force  of  the  Daniell  cell,  involving  the  above 
combination,  is  1.096  volts. 

Measurement  of  the  Difference  of  Potential  between  a  Metal 
and  a  Solution.  The  difference  of  potential  between  a  metal 
and  a  solution  of  one  of  its  salts  is  easily  determined  by  means  of 
the  calomel  electrode.  For  example,  in  order  to  determine  the 
potential  of  zinc  against  a  molar  solution  of  zinc  sulphate,  the 
electromotive  force  TT,  of  the  combination 

Zn  -  m  ZnS04||  m  KC1,  HgCl  -  Hg, 

(cal.  electrode) 

is  measured  and  found  to  be  1.08  volts,  the  mercury  being  the 
positive  terminal  of  the  cell.  Applying  the  Nernst  equation, 

RT,       P!      RT,       P2 

7r  =  7ri-T2=_loge---_loge-, 

in  which  PI  and  pi  denote  the  solution  pressure  and  the  osmotic 
pressure  of  the  zinc  ions,  and  P2  and  pz  denote  the  solution  pressure 
and  the  osmotic  pressure  of  the  mercury  ions,  we  have 


or 


1.08  =        log,       -(-  0.56), 


loge    1  J  =  1.08  -  0.56  =  0.52  volt. 
2V          pi 


That  is,  the  zinc  electrode  is  negative  against  a  molar  solution  of 
zinc  sulphate,  the  difference  of  potential  being  0.52  volt.    As  an 


ELECTROMOTIVE  FORCE  415 

example  of  a  cell  in  which  the  mercury  of  the  calomel  electrode 
is  the  negative  terminal  of  the  cell,  we  may  take  the  following 
combination:  — 

Cu  -  m  CuS04  H  m  KC1,  HgCl  -  Hg. 

The  electromotive  force  of  this  cell  is  0.025  volt.  Since  the 
current  flows  from  the  copper  to  the  mercury,  we  have 


,  =  0.025  =  -0.56  -     p  loge      , 
or 

•pm  p 

££  ioge  £J  =  _  o.025  -  0.56  =  -0.585  volt. 
Zf  pi 

That  is,  the  copper  electrode  is  positive  against  a  molar  solution 
of  copper  sulphate.  From  the  above  results  it  is  possible  to  cal- 
culate the  electromotive  force  of  the  combination 

Zn  -  m  ZnS04  1|  m  CuS04  -  Cu. 
Since,  according  to  Nernst's  equation 

RT.       Pi      RT  .       P2 


where  PI  and  pi  refer  to  the  zinc,  and  P2  and  p2  refer  to  the  copper, 
we  have 

TT  =  0.52  -  (-  0.585)  =  1.105  volts. 

Concentration  Elements.  We  now  proceed  to  consider  cells 
hi  which  the  electromotive  force  depends  primarily  on  differences 
in  concentration,  —  the  so-called  "concentration  elements." 

Concentration  elements  may  be  conveniently  divided  into  two 
classes:  (a)  elements  in  which  the  electrodes  are  of  different  concen- 
trations,  and  (b)  elements  in  which  the  solutions  are  of  different 
concentrations. 

(a)  Elements  in  which  the  Electrodes  are  of  Different  Concentra- 
tions. (Amalgams  and  Alloys.)  If  in  the  equation 

RT.       P!      RT 


416 


THEORETICAL  CHEMISTRY 


PI  =  Pz,  as  is  the  case  when  the  ionic  concentrations  of  the  two 
solutions  are  identical,  then  we  have 

RT,      P1 


where  PI  and  P%  are  the  respective  solution  pressures  of  the  metal 
dissolved  in  the  electrodes.  If  the  amalgams  are  dilute,  the 
osmotic  pressure  of  the  dissolved  metal  will  be  proportional  to  the 
solution  pressure  of  the  electrode,  and  since  osmotic  pressure 
is  proportional  to  concentration,  we  may  replace  PI  and  P2  in  the 
above  formula  by  the  proportional  terms,  c\  and  C2,  the  respec- 
tive concentrations  of  the  metal  hi  the  two  electrodes.  Hence, 
we  have 

RT 


The  accuracy  of  this  equation  has  been  fully  established  by  the 
experiments  of  Meyer,*  and  Richards  and  Forbes.  f 

Meyer's  results  for  zinc  amalgams  in  solutions  of  zinc  sulphate 
are  given  in  the  accompanying  table. 


,1  T' 

degrees. 

Cl 

Cl 

it  (obs.). 

T  (calc.). 

284.6 

0.003366 

0.00011305 

0.0419 

0.0416 

291.0 

0.003366 

0.00011305 

0.0433 

0.0425 

285.4 

0.002280 

0.0000608 

0.0474 

0.0445 

333.0 

0.002280 

0.0000608 

0.0520 

0.0519 

The  agreement  between  the  observed  and  calculated  values  of  w 
is  all  that  can  be  desired.  That  the  above  formula  holds  for 
zinc  amalgams  may  be  considered  as  a  proof  of  the  fact  that  the 
zinc  dissolves  in  the  mercury  as  monatomic  molecules.  Thus, 
suppose  the  zinc  to  be  present  in  the  mercury  in  the  form  of  dia- 
tomic molecules;  then  while  the  electrical  energy  would  be  equal 

*  Zett.  phys.  Chem.,  7,  477  (1891). 

t  Publication  of  the  Carnegie  Institution,  No.  56. 


ELECTROMOTIVE  FORCE  417 

to  2  Fir,  the  osmotic  work  required  to  develop  this  energy  would 
bep:  RT  loge  -  ,  hence  we  should  have 

6  Cz 

1  RT 


or  the  calculated  value  of  the  electromotive  force  would  be  just 
one-half  of  the  observed  value.  The  mercury  in  the  amalgam 
has  been  shown  to  exert  no  effect  upon  the  electromotive  force  of 
the  cell  so  long  as  the  dissolved  metal  has  the  greater  potential. 

(b)  Elements  in  which  the  Solutions  are  of  Different  Concentra- 
tions. In  this  type  of  cell  we  have  two  electrodes  of  the  same 
metal  immersed  in  solutions  of  different  ionic  concentrations  of 
the  metal.  Hence,  we  may  put  PI  =  P2  in  the  equation 

RT,      Pl      RT,      P2 

*=^log<^ 

which  then  takes  the  form 

RT 

7T=  —  =• 

nF 

Since  osmotic  pressure  is  proportional  to  concentration,  pi  and 
pz  may  be  replaced  by  the  proportional  terms  Ci  and  Cz,  and  the 
foregoing  equation  becomes 

RT 


or 

RT  .      mien 


where  m\  and  mz  are  the  molar  concentrations  of  the  two  solutions 
and  ai  and  az  are  the  corresponding  degrees  of  ionization.  As  an 
example  of  a  concentration  element  of  this  class  we  may  take  the 
following  :  — 

Ag  -  0.01  m  AgN03  1|  0.1  m  AgN03  -  Ag. 


418  THEORETICAL  CHEMISTRY 

The  degrees  of  ionization  of  the  two  solutions  at  18°  are  as  follows: 
—  for  0.01  m  AgNO3,  a  =  0.93,  and  for  0.1  m  AgN03,  a  =  0.81. 
Substituting  in  the  equation 

IT  =0.058  log 
6 

we  have 


The  value  of^-n-  found  by  direct  experiment  is  0.055  volt. 

In  the  example  just  given,  the  electrodes  are  reversible  with 
respect  to  the  positive  ion  of  the  electrolyte.  Such  electrodes  are 
known  as  electrodes  of  the  first  type.  It  is  also  possible  to  construct 
cells  with  electrodes  which  are  reversible  with  respect  to  the  nega- 
tive ion  of  the  electrolyte.  These  are  termed  electrodes  of  the  second 
type.  The  calomel  electrode  is  an  example  of  an  electrode  of  this 
latter  type.  If  positive  electricity  passes  from  the  metal  to  the 
solution,  the  mercury  combines  with  the  Cl'  ions  forming  mercu- 
rous  chloride,  and  if  positive  electricity  passes  in  the  reverse 
direction,  chlorine  dissolves  and  mercurous  chloride  is  formed. 
In  other  words  the  electrode  behaves  like  a  chlorine  electrode, 
giving  up  or  absorbing  the  element  according  to  the  direction  of 
the  current.  A  typical  combination  involving  an  electrode  of 
the  second  type  is  the  following  :  — 

Ag  -  0.1  m  AgN03  -  KNO3  -  0.1  m  KC1,  AgCl  -  Ag. 

This  particular  combination  was  studied  by  Goodwin  *  with  a 
view  to  determining  the  solubility  of  silver  chloride.  If  we  assume 
a  saturated  solution  of  silver  chloride  to  be  completely  ionized, 
then  the  solubility  product  will  be 

CAg«  X  Cd'  =  S. 

Since  the  concentrations  of  the  two  ions,  Ag"  and  Cl',  are  equal, 
it  follows  that  Vs  will  be  equal  to  the  solubility  of  the  silver 

*  Zeit.  phys.  Chem.,  13,  577,  (1894). 


ELECTROMOTIVE  FORCE  419 

chloride.     The  electromotive  force  of  a  concentration  cell  at  25° 
is  given  by  the  equation 

0.0595  . 

7T  =  -  log 
n 

or 

W 

8 


0.0595 

The  value  of  TT  at  25°  for  the  above  cell  was  found  to  be  0.450  volt. 
The  degrees  of  ionization  of  the  two  electrolytes  are  as  follows:  — 
for  0.1  molar  AgNO3,  a  =  0.82,  and  for  0.1  molar  KC1,  a  =  0.85. 
Substituting  in  the  preceding  equation,  we  obtain 

0.1  X  0.82  =  0.450 

C2  0.0595' 

therefore, 

ft  =  1.94  X  10~9. 

Or,  1.94  X  10~9  is  the  concentration  of  the  Ag*  ion  in  mols  per 
liter  in  a  0.1  molar  potassium  chloride  solution  of  silver  chloride. 
Hence  the  solubility  product  s,  will  be 

8  =  1.94  X  10~9  X  0.85  =  1.64  X  lO"10, 
and 

Vs  =  1.28  X  10~5; 

that  is,  the  solubility  of  silver  chloride  in  a  saturated  aqueous 
solution  is  1.28  X  10~5  mol  per  liter  at  25°. 

The  Difference  of  Potential  at  the  Junction  of  the  Solutions 
of  Two  Electrolytes.  Thus  far  we  have  not  taken  into  consider- 
ation the  potential  differences  which  may  be  established  at  the 
junction  of  two  solutions.  Nernst  *  has  shown  that  in  many  cases 
it  is  possible  to  calculate  these  differences  of  potential  by  means 
of  his  osmotic  theory  of  the  origin  of  electromotive  force,  and  the 
values  obtained  are  in  close  agreement  with  the  results  of  experi- 
ment. Let  us  imagine  that  two  solutions  of  hydrochloric  acid  of 
different  concentrations  are  brought  together  so  as  to  avoid  mix- 
ing, the  acid  in  each  solution  being  highly  ionized.  The  hydro- 
gen and  chlorine  ions  will  diffuse  independently,  and  since  the 
*  Zeit.  phys.  Chem.,  4,  129  (1889). 


420  THEORETICAL  CHEMISTRY 

former  move  with  the  greater  velocity,  the  more  dilute  solution 
will  soon  contain  an  excess  of  H*  ions  and  the  more  concentrated 
solution  an  excess  of  Cl'  ions.  The  more  dilute  solution  will  be- 
come positively  charged  owing  to  the  presence  of  an  excess  of  H" 
ions,  while  the  more  concentrated  solution  will  acquire  a  negative 
charge  due  to  the  presence  of  an  excess  of  Cl'  ions.  The  accumu- 
lation of  positive  electricity,  however,  will  retard  the  velocity  of 
the  H"  ions  and  accelerate  the  velocity  of  the  Cl'  ions,  so  that 
ultimately  the  two  ions  will  move  with  the  same  velocity.  The 
difference  of  potential  produced  in  this  way  will  cease  to  exist 
when  the  two  solutions  have  acquired  the  same  concentration. 

In  general,  it  may  be  stated  that  the  difference  of  potential  set 
up  at  the  junction  of  the  two  solutions  is  caused  by  the  differ- 
ence in  the  rates  of  migration  of  the  two  ions,  the  more  dilute 
solution  acquiring  a  charge  corresponding  to  that  of  the  faster 
moving  ion.  The  formula  expressing  the  difference  of  potential 
TT,  between  two  solutions  containing  the  same  univalent  ions  in 
different  concentrations,  may  be  derived  in  the  following  manner: 
—  Let  u  and  v  be  the  migration  velocities  of  the  cation  and  the 
anion  respectively,  and  let  pi  be  the  osmotic  pressure  of  the  ions 
hi  the  concentrated  solution  and  p2  the  osmotic  pressure  of  the 
ions  in  the  dilute  solution.  Now  let  1  faraday  of  electricity  pass 
through  the  two  solutions,  the  current  entering  on  the  concen- 

u 
trated  side;    then  — ^—  gram-equivalents  of  positive  ions  will 

migrate  from  the  concentrated  to  the  dilute  solution,  while  — -:— 

gram-equivalents  of  negative  ions  will  migrate  from  the  dilute  to 
the  concentrated  solution.  The  maximum  work  done  by  the 
process  will  be 

— ^—  RT loge  ^ V—  RT  loge  -  =  ^f-^  RT loge  ^- 

u  +  v  pz      u  +  v  p2      "'  '   - 

have 


The  corresponding  electrical  energy  developed  is  irF,  hence  we 
have 


u~  v .  "*  w  £1 
u  +  v     F   10g>2 


ELECTROMOTIVE  FORCE  421 

or,  since  osmotic  pressure  is  proportional  to  concentration,  we 
may  substitute  Ci  and  C2  for  pi  and  p2  in  the  preceding  equation, 
and  obtain  the  following  expression  for  the  electromotive  force 
at  the  junction  of  the  two  solutions:  — 

u-vRT,       ci 


As  an  example  of  the  use  of  the  above  formula,  we  may  take  the 
calculation  of  the  electromotive  force  of  the  following  combina- 
tion 

Ag  -  0.1  m  AgN03  -  0.01  m  AgNO3  -  Ag. 

(a)  (b)  (c) 

Taking  the  junctions  (a),  (b),  and  (c)  in  order,  we  obtain 
RT        C  .  u  -  v    RT  ,      a      RT  ,      C 


u-vRT,         ci      RT 


Assuming  the  temperature  to  be  17°  and  passing  to  Briggsian 
logarithms,  we  have 

TT  =  11?-?  0.058  log  ^- 

U  +  V  &C2 

The  transport  number  for  the  anion  of  silver  nitrate  is  0.522, 
while  ci  =  mxxi  =  0.1  X  0.82  =  0.082,  and  &  =  ra2a2  =  0.01  X 
0.94  =  0.0094.  Hence, 

TT  =  -2  X  0.522  X  0.058  X 

or 

TT  =  -0.057  volt. 

The  value  found  by  direct  experiment  is  —  0.055  volt. 

Oxidation  and  Reduction  Elements.  When  a  dissolved  sub- 
stance passes  from  a  lower  to  a  higher  state  of  oxidation,  the 
change  in  the  positive  ion  may  be  considered  as  due  to  an  increase 
in  the  number  of  electrical  charges  on  the  ion;  thus,  when  a  ferrous 


422  THEORETICAL  CHEMISTRY 

salt  is  oxidized  to  the  ferric  state,  the  change  may  be  represented 
by  the  equation 

Fe"  +  (+)->  Fe.- 

Similarly,  the  reduction  of  a  ferric  salt  to  the  corresponding  ferrous 
salt  may  be  represented  by  the  reverse  equation,  or 

Fe~-»Fe"  +  (+). 

The  formation  of  zinc  ions  from  metallic  zinc  may  be  considered 
as  an  oxidation,  and  may  be  represented  by  the  equation 


The  formation  of  negative  ions  from  a  non-metallic  element  may 
be  considered  as  a  reduction,  as  for  example,  the  change  of  potas- 
sium ferricyanide  to  potassium  ferrocyanide,  which  may  be  rep- 
resented by  the  equation 

Fe(CN)6'"  +  (-)->  Fe(CN)6"". 

The  foregoing  considerations  lead  to  the  following  definition  of 
the  terms  oxidation  and  reduction:  Oxidation  is  the  process  in 
which  a  substance  takes  up  positive  or  parts  with  negative  charges, 
and  reduction  is  the  process  in  which  a  substance  takes  up  negative 
or  parts  with  positive  charges. 

The  potential  difference  between  a  metal  and  a  solution  of  one 
of  its  salts  is  a  measure  of  the  tendency  of  the  metal  to  form  ions, 
or  in  other  words  is  the  criterion  of  its  tendency  towards  oxidation. 
The  tendency  of  an  ion  in  a  lower  state  of  oxidation  to  pass  over 
into  a  higher  state  of  oxidation  may  be  determined  by  means  of 
electromotive  force  measurements. 

A  typical  oxidation  cell  is  the  following  :  — 

Pt  -  Fe"  ||  Fe"'  -  Pt. 

If  the  platinum  electrodes  are  connected  to  a  lead  accumulator, 
and  a  current  is  passed  in  the  direction  of  the  arrow 

Pt-Fe"||Fe--Pt, 


ELECTROMOTIVE  FORCE  423 

the  ferrous  ions  on  one  side  of  the  element  will  all  be  oxidized  to 
the  ferric  state,  while  the  ferric  ions  on  the  other  side  will  all  be 
reduced  to  the  ferrous  state,  thus: 

Pt  -  Fe'"  ||  Fe"  -  Pt. 

If  now  the  connection  with  the  accumulator  is  broken  and  the  two 
platinum  electrodes  are  connected  with  a  wire,  the  following  proc- 
ess will  take  place  :  —  ferric  ions  will  give  up  positive  charges  to 
one  electrode,  thereby  being  reduced  to  ferrous  ions,  while  the 
charges  given  up  pass  along  the  wire  to  the  other  electrode,  there 
converting  some  ferrous  ions  to  ferric  ions.  This  process  will 
continue  until  an  equilibrium  is  established,  in  which  the  ratio 
Fe"  :  Fe*"  will  be  the  same  in  both  solutions.  The  electromotive 
force  corresponding  to  this  change  gives  a  measure  of  the  tendency 
of  the  ions  to  undergo  oxidation,  and  is  directly  proportional  to 
the  ionic  concentrations. 

The  following  modification  of  the  Nernst  equation  gives  the 
relationship  between  ionic  concentrations  and  the  resulting  elec- 
tromotive force:  — 


j>,,       Co 
Ci)  =  jP+_loge-, 

where  7r(Co->c.)  is  the  electromotive  force  produced  by  the  passage 
from  the  lower,  or  "ous"  state  of  oxidation  to  the  higher,  or  "ic" 
state  of  oxidation;  P  is  the  potential  when  the  concentrations  of 
the  "ous"  and  "ic"  ions  are  equal,  c0  and  ct-  are  the  ionic  con- 
centrations of  the  lower  and  higher  states  of  oxidation  respectively, 
and  n  is  the  difference  in  valence  of  the  two  kinds  of  ions.  A 
number  of  oxidation  and  reduction  elements  have  been  studied 
by  Bancroft.* 

Electrolytic  Solution  Pressure.  The  electrolytic  solution 
pressure  of  a  metal  expressed  in  atmospheres,  can  be  calculated 
from  the  difference  of  potential  between  the  metal  and  a  solution 
of  one  of  its  salts. 

*  Zeit.  phys.  Chem.,  10,  387  (1892). 


424  THEORETICAL  CHEMISTRY 

Thus,  solving  the  equation 

RT 


for  P,  and  passing  to  Briggsian  logarithms,  we  have,  at  17°, 

irn 


logP 


-flogp 


290 
22.4  a          atmos- 


0.085 

Employing   molar   solutions   and  taking  p 

pheres,  Neumann  *  calculated  the  electrolytic  solution  pressures 
in  atmospheres  of  the  metals  given  in  the  accompanying  table. 


Metal. 

Solution  Pressure, 
(atmos.). 

Magnesium  ...                    .    . 

10      X  1043 

Zinc 

9.9X1018 

Cadmium 

2.7X106 

Thallium 

7.7X102 

Iron 

1.2X104 

Cobalt 

1.9X10° 

Nickel.. 

1.3X10° 

Lead 

1  1X10~3 

Hydrogen 

9  9xlO~4 

Copper 

4  8X10-20 

Mercury 

1  1X10-16 

Silver  

2.3X10-17 

Palladium 

1  5X10-36 

Heat  of  lonization.  If  the  difference  of  potential  between  a 
metal  and  the  solution  of  one  of  its  salts  is  known,  together  with 
its  temperature  coefficient,  it  is  possible  to  calculate  the  heat  of 
ionization  of  the  metal  by  means  of  the  Gibbs-Helmholtz  equation, 

W.±  +  T*L. 

nF^      dT 

Solving  the  equation  for  Q,  the  heat  evolved  when  one  mol  of  ions 
is  formed  at  the  electrode,  we  have 

<*•(*-*)&'*• 

*  Zeit.  phys.  Chem.,  14,  225  (1894). 


ELECTROMOTIVE  FORCE 


425 


For  example,  the  potential  of  zinc  against  a  molar  solution  of  zinc 
chloride  is  —  0.497  volt  at  25°,  and  the  temperature  coefficient  of 
electromotive  force  is  0.000664  volt  per  degree.  Substituting  in 
the  equation,  we  have 

Q  =  [-  0.497  -  (273  +  25)  X  0.000664]  (2  X  96,540  X  0.2394) 
or  Q  =  32,120  calories. 
That  is,  the  heat  of  the  reaction 

Zn-f2(+)->Zn" 

is  32,120  calories  per  mol  of  zinc. 

Gas  Cells.  It  is  interesting  to  note  that  gases  may  function 
as  electrodes  in  much  the  same  way  as  metals  or  amalgams.  Gas 
electrodes  are  usually  prepared  by  partially 
immersing  strips  of  platinized  platinum  in 
a  solution  of  a  suitable  electrolyte,  and 
bubbling  the  gas  through  the  solution  until 
a  constant  difference  of  potential  is  estab- 
lished between  it  and  the  electrode.  A  very 
satisfactory  form  of  gas  electrode  is  shown 
in  Fig.  99.  Reference  has  already  been  made 
to  the  hydrogen  electrode  in  connection 
with  the  measurement  of  single  electrode 
potentials. 

This  electrode  is  completely  reversible 
and  behaves  like  a  plate  of  metallic  hydro- 
gen, the  reaction  at  the  electrode  being 
represented  by  the  equation 


The  amount  of  energy  developed  by  the 

passage  of  a  certain  quantity  of  gas  into 

the  ionic^state  is  precisely  the  quantity  nec- 

essary and  sufficient  to  produce  the  reverse  Fig.  99. 

action. 

This  being  true,  the  metal  of  the  electrode  can  exert  no  influence 
upon  the  electromotive  force.  A  hydrogen  concentration  cell 
can  be  formed  by  connecting  two  hydrogen  electrodes,  containing 


426  THEORETICAL  CHEMISTRY 

the  gas  at  different  pressures,  through  an  intermediate  electrolyte. 
The  direction  of  the  current  is  such  that  the  pressures  on  the  two 
sides  of  the  cell  tend  to  become  equal,  molecular  hydrogen  being 
ionized  on  the  high  pressure  side,  and  ionized  hydrogen  being  dis- 
charged on  the  low  pressure  side. 

The  electromotive  force  of  such  a  cell  can  be  calculated  by  means 
of  the  Nernst  equation.  Let  us  consider  a  cell  composed  of  two 
hydrogen  electrodes,  each  at  atmospheric  pressure,  the  H"  ion 
concentration  in  each  being  ci,  then 

RT.      Ci     RT 


where  Ci  is  the  molecular  concentration  of  the  hydrogen  dissolved 
in  the  platinum  at  atmospheric  pressure  p\.  Since  the  hydrogen 
is  present  in  the  form  of  diatomic  molecules,  n  =  2. 

If  now  the  pressure  of  the  gas  at  one  electrode  be  increased  to 
p2)  and  the  corresponding  molecular  concentration  of  the  hydro- 
gen in  the  electrode  be  C2,  then  we  shall  have 

RJ.  •,       GI      RT  ,      C-2 
*=2FloS-«-2Flos«' 
or  since 

Ci  :C2  ::pi  :p2, 


Equation  (1)  applies  equally  well  to  cells  in  which  two  different 
gases  are  employed.  If  solution  pressures  be  used  in  the  calcu- 
lation of  the  electromotive  force  of  a  gas  cell,  equation  (1)  becomes 

RT  .       PI  ,_. 

-,  (2) 


where  PI  and  P2  are  the  solution  pressures  of  the  two  gases.  Since 
the  values  of  TT  obtained  by  equations  (1)  and  (2)  must  be  equal, 
we  may  write 

RT  .       pi      RT.       Pi 


ELECTROMOTIVE  FORCE  427 

and 

therefore 


That  is,  the  raticTof  the  actual  gas  pressures  is  equal  to  the  ratio 
of  the  squares  of  the  corresponding  solution  pressures. 

lonization  of  Water.  An  important  application  of  the  gas 
cell  is  its  use  in  determining  the  degree  of  ionization  of  water. 
If  we  measure  the  electromotive  force  of  the  cell 

PtHl  -  0.1  m  NaOH  -  0.1  m  HC1  -  PtHl, 

and  determine  the  concentration  of  the  H"  ions  on  one  side  and  the 
concentration  of  the  OH'  ions  on  the  other  side,  we  can  calculate 
the  concentration  of  the  H"  ions  in  the  sodium  hydroxide  solution. 
The  reaction  which  produces  the  current  is  represented  by  the 
equation 

NaOH  +  HC1  =  NaCl  +  H2O, 

or  more  correctly 

H'  +  OH'  =  H20. 

At  25°  the  electromotive  force  of  the  above  cell  is  0.646  volt. 
At  the  junction  of  the  two  solutions  an  electromotive  force  of 
0.0468  volt  is  set  up;  hence  the  true  electromotive  force  of  the 
cell  is  0.646  +  0.0468  =  0.6928  volt.  The  degree  of  ionization 
of  a  0.1  molar  solution  of  hydrochloric  acid  is  a\  =  0.924,  and 
the  degree  of  ionization  of  a  0.1  molar  solution  of  sodium 
hydroxide  is  a2  =  0.847.  Introducing  these  values  into  the 
Nernst  equation 

RT 

7r  =  W 
we  have 

0.6928  =  0.0595  log0'1  X  °-924. 

Solving  this  equation  for  02,  the  concentration  of  the  H*  ions 
in  the  sodium  hydroxide  solution,  we  find  c%  to  be  equal  to 
1.66  X  10~13. 


428  THEORETICAL  CHEMISTRY 

Therefore 

s  =  cH-  X  COH'  =  1.66  X  10~13  X  0.1  X  0.847  =  1.406  X  10~14, 

and  

CH-  =  COH'  =  VJ  =  Vl.406  X  10-14  =  1.187  X  10~7. 

This  value  is  in  excellent  agreement  with  the  values  obtained  from 
measurements  of  the  electrical  conductance  of  water  and  the  speed 
of  hydrolysis  of  esters.  The  values  of  the  degree  of  ionization  of 
water  at  25°,  as  determined  by  these  three  methods  are  as  fol- 
lows :  — 

Electrical  conductance  of  pure  water 1 .05  X  10~7 

Velocity  of  hydrolysis  of  methyl  Acetate. . .    1.2    X  10~7 
E.M.F.  of  hydrogen-oxygen  cell 1.18  X  10~7 

When  we  consider  the  exceedingly  small  extent  to  which  water 
is  ionized,  the  close  agreement  between  these  results  is  most  satis- 
factory. The  correctness  of  these  figures  can  be  further  checked 
by  taking  the  values  of  the  degree  of  ionization  of  water  at  two 
temperatures,  and  calculating  the  heat  of  the  reaction 

H*  +  OH'  =  H20, 

by  means  of  Van't  Hoff's  isochore  equation. 

Thus,  according  to  Kohlrausch  the  degree  of  ionization  of  water 
is  0.35  X  10~7  at  0°,  and  2.48  X  10~7  at  50°.  Introducing  these 
values  into  the  equation 

R  X  2.306  (log  K2  -  log  KJ  Ttf* 

<? — Tfi If —  —  > 

1  1  —  lz 

and  solving  for  q,  we  obtain  13,810  calories,  a  value  agreeing  well 
with  that  found  by  the  direct  measurement  of  the  heat  of  neutral- 
ization of  completely  ionized  acids  and  bases,  viz.,  13,700  calories. 
Storage  Cells  or  Accumulators.  Storage  cells  or  accumulators, 
as  the  name  implies,  are  devices  for  the  storage  of  electrical  energy 
in  the  form  of  chemical  energy.  Any  reversible  cell  may  be 
employed  as  an  accumulator.  Thus,  the  oxygen-hydrogen  cell 

Ptoj  —  Solution  of  Sulphuric  Acid  —  Ptna 

may  be  used  as  an  accumulator,  if  the  gases  resulting  from  the 
electrolysis  of  water  are  collected  at  the  electrodes  and  then  used 


ELECTROMOTIVE  FORCE  429 

to  produce  a  current.  In  actual  work,  the  lead  accumulator  is 
used  almost  exclusively.  If  two  lead  plates  are  immersed  in  a 
20  per  cent  solution  of  sulphuric  acid,  a  minute  amount  of  lead 
sulphate  will  be  formed  on  the  surface  of  each  plate.  If  now  a 
current  of  electricity  is  passed  through  the  solution,  the  lead 
sulphate  on  the  cathode  will  undergo  reduction  to  metallic  lead, 
and  the  lead  sulphate  on  the  anode  will  be  oxidized  to  lead  peroxide. 
In  this  way  we  form  the  cell 

Pb  -  20  per  cent  Sol.  H2S04  -  Pb02, 

the  electromotive  force  of  which  is  about  2  volts.  The  amounts 
of  lead  and  lead  peroxide  produced  in  this  way  are  so  small  that 
the  cell  can  only  furnish  a  very  small  amount  of  electrical  energy. 
In  order  to  increase  its  capacity,  the  electrodes  should  be  given 
as  large  an  amount  of  surface  as  possible.  This  may  be  brought 
about  by  the  method  of  Plante,  in  which  the  solution  is  elec- 
trolyzed  first  in  one  direction  and  then  in  the  other,  thus  causing 
the  plates  to  become  spongy;  or  by  the  method  of  Faure,  in  which 
a  lead  "grid"  is  charged  with  a  paste  of  lead  oxide  and  red  lead, 
and  is  then  introduced  into  the  solution  and  the  current  passed 
until  we  obtain  spongy  lead  at  the  cathode  and  lead  peroxide  at  the 
anode.  If  now  the  charging  circuit  is  broken  and  the  two  elec- 
trodes are  connected  by  a  wire,  a  current  will  flow  from  the  peroxide 
plate  to  the  lead  plate,  lead  sulphate  being  slowly  formed  at  each. 
In  charging  the  accumulator,  the  lead  sulphate  on  the  negative 
electrode  is  reduced  to  metallic  lead,  the  reaction  being  represented 
by  the  following  equation :  — 

PbSO4  +  2  (-)  =  Pb"  +  SO/'. 

At  the  positive  electrode  SO/'  ions  are  liberated  and  they  react 
with  the  lead  sulphate  and  the  water  of  the  electrolyte  in  the 
following  manner :  — 

PbS04  +  2  H20  +  S04"  +  2  (+)  =  Pb02  +  4  IT  +  2  S04". 

When  the  cell  is  discharged  S04"  ions  are  liberated  at  the  lead 
plate  forming  lead  sulphate,  as  shown  by  the  equation 

Pb  +  SO/'  +  2  (+)  =  PbSO4. 


430  THEORETICAL  CHEMISTRY 

At  the  peroxide  plate  H*  ions  are  discharged  and,  in  the  presence 
of  the  electrolyte,  convert  the  lead  peroxide  into  lead  sulphate, 
according  to  the  equation 

Pb02  +  2  IT  +  H2S04  +  2  (-)  =  PbS04  +  2  H20. 

Combining  the  foregoing  equations,  we  obtain  the  following  single 
equation  summarizing  the  chemical  changes  involved  in  the  pro- 
duction of  the  current:  — 

Pb02  +  Pb  +  2  H2S04  <=±  2  PbS04  +  2  H20. 

The  upper  arrow  represents  the  reaction  on  discharging,  while 
the  lower  arrow  represents  the  reaction  on  charging. 

The  electromotive  force  of  the  storage  cell  is  approximately 
2  volts.  It  is  not  completely  reversible,  but  under  favorable 
conditions  its  efficiency  is  about  90  per  cent;  that  is,  90  per  cent 
of  the  electrical  energy  supplied  to  it  in  charging  can  be  recov- 
ered on  discharging.* 

PROBLEMS. 

1.  Calculate  the  heat  of  amalgamation  of  cadmium  at  0°  from  the 
following  data :  —  the  electromotive  force  of  a  cell  made  up  of  a  1-per  cent 
cadmium  amalgam  in  a  solution  of  cadmium  sulphate  is  0.06836  volt  at 
0°,  and  0.0735  volt  at  24° .45.  Ans.  510  calories  per  mol  of  Cd. 

2.  The  electromotive  force  of  the  cell 

Pb  -  0.01  m  Pb(N03)2  -  sat.  NH4N03  -  m  KC1,  HgCl  -  Hg 
is  -  0.469  volt  at  25°.    The  lead  nitrate  is  62  per  cent  ionized.    What 
is  the  potential  of  lead  against  a  solution  containing  1  mol  of  Pb.  ions 
per  liter,  referred  to  the  calomel  electrodes  as  zero? 

Ans.   -  0.405  volt. 

3.  Calculate  the  electromotive  force  of  the  cell 

Cu  amalgam  (a)  —  Solution  CuSO4  —  Cu  amalgam  (b), 
at  20°.8,  having  given  that  the  concentrations  of  the  amalgams  (a)  and 
(b)  are  0.0004472  and  0.00016645  respectively. 

Ans.   0.0125  volt. 

*  For  a  thorough  treatment  of  the  theory  of  the  lead  accumulator  the 
student  is  recommended  to  consult  "Die  Theorie  des  Bleiaccumulators, " 
by  F.  Dolezalek. 

For  a  detailed  account  of  the  primary  cells  in  common  use  the  reader  will 
find  Carhart's  "Primary  Batteries"  most  satisfactory. 


ELECTROMOTIVE  FORCE  431 

4.  Calculate  the  electromotive  force  of  the  cell 

Cu  -  m  CuS04  -  0.01  m  CuS04  -  Cu, 

at  25°,  having  given  the  following  values  for  the  degree  of  ionization  of 
the  two  solutions:  —  for  m  copper  sulphate,  a  =  0.21,  and  for  0.01  m 
copper  sulphate,  a  =  0.61.  The  electromotive  force  at  the  junction  of 
the  two  solutions  may  be  neglected.  Arts.  0.0458  volt. 

5.  At  25°  the  electromotive  force  of  the  cell 

Zn  -  0.5  m  ZnS04  -  0.05  m  ZnS04  -  Zn 

is  0.018  volt.  Neglecting  the  potential  developed  at  the  junction  of  the 
solutions,  and  assuming  the  dilute  solution  of  zinc  sulphate  to  be  ionized 
to  the  extent  of  35  per  cent,  find  the  degree  of  ionization  of  the  concen- 
trated solution.  Ans.  0.142. 

6.  What  is  the  electromotive  force  of  the  cell 

Zn  -  0.1  m  ZnS04  -  0.01  m  ZnS04  -  Zn, 

at  18°?    For  ZnS04,  — V—  =  0.601,  for  0.1  molar  ZnS04,  «  =  0.39,  and 
u  +  v 

for  0.01  molar  ZnS04,  a  =  0.63.  Ans.  0.074  volt. 

7.  The  electromotive  force  of  the  cell 

Ag  -  0.001  m  AgN03  -  m  KN03  -  m  KI,  Agl  -  Ag 

is  0.22  volt  at  18°.  A  molar  solution  of  KI  is  78  per  cent  ionized,  and  a 
0.001  molar  solution  of  AgN03  is  98  per  cent  ionized.  Calculate  the  solu- 
bility of  Agl.  Ans.  1.11  X  10~4  mols  per  liter. 

8.  The  potential  of  zinc  against  a  solution  containing  one  mol  of 
Zn  ions  is  0.493  volt,  at  18°.    Assuming  complete  dissociation,  calculate 
the  solution  pressure  of  zinc  in  atmospheres. 

9.  The  electromotive  force  of  the  Daniell  cell 

Cu  -  CuS04  -  ZnS04  -  Zn 

is  1.0960  volt  at  0°,  and  1.0961  volt  at  3°.  Calculate  the  heat  of  the  reac- 
tion taking  place  in  the  cell.  Ans.  63,390  calories. 

10.  Calculate  the  electromotive  force  of  the  cell 

Ptn2  -  0.1  m  KOH  -  m  HC1  -  PtHil 

at  25°,  having  given  that  0.1  molar  KOH  is  85  per  cent  ionized  and  molar 
HC1  is  70  per  cent  ionized.  Ans.  0.757  volt. 


CHAPTER  XVIII. 

ELECTROLYSIS  AND  POLARIZATION. 

Polarization.  If  a  difference  of  potential  of  about  1  volt  is 
applied  to  two  platinum  electrodes  immersed  in  a  concentrated 
solution  of  hydrochloric  acid,  it  will  be  found  that  the  current 
which  passes  at  first,  steadily  diminishes  and  ultimately  becomes 
zero.  The  cessation  of  the  current  has  been  shown  to  be  due  to 
the  accumulation  of  hydrogen  on  the  cathode  and  chlorine  on  the 
anode,  these  two  gases  setting  up  an  opposing  electromotive  force 
called  the  electromotive  force  of  polarization. 

If  in  the  above  case  the  applied  electromotive  force  is  increased 
to  1.5  volts,  the  counter  electromotive  force  is  no  longer  sufficient 
to  reduce  the  current  to  zero.  In  fact,  at  any  voltage  above  1.35 
volts  a  continuous  current  passes;  this  is  termed  the  decomposition 
potential  of  hydrochloric  acid. 

Above  the  decomposition  potential,  the  current  C  may  be 
calculated  by  means  of  the  formula 

E  -  e  =  CR, 

where  E  is  the  applied  electromotive  force,  e  the  counter  electro- 
motive force,  and  R  the  resistance  of  the  electrolyte. 

As  the  applied  electromotive  force  and  the  current  increase,  the 
polarization  increases,  since  the  gases  are  liberated  under  a  pres- 
sure greater  than  that  of  the  atmosphere;  but  since  the  gases 
escape  from  the  solution  the  value  of  e  can  never  become  equal 
to  E.  The  decomposition  potential  of  an  electrolyte  can  be 
determined  in  two  different  ways,  viz.:  (1)  by  gradually  raising 
the  applied  electromotive  force  E  until  it  exceeds  e,  when  the 
current  will  suddenly  increase;  or  (2)  by  charging  the  electrodes 
up  to  atmospheric  pressure  by  means  of  an  electromotive  force 
greater  than  e,  and  then  breaking  the  external  circuit  and  measur- 
ing the  counter  electromotive  force. 

432 


ELECTROLYSIS  AND  POLARIZATION 


433 


The  arrangement  of  apparatus  for  the  measurement  of  the 
electromotive  force  of  polarization,  as  suggested  by  Le  Blanc,* 
is  indicated  in  Fig.  100.  A  is  the  cell  in  which  polarization 
occurs,  B  is  the  source  of  external  electromotive  force,  C  is  a  capil- 
lary electrometer,  D  is  a  source  of  variable  potential,  and  E  is  one 
prong  of  an  electrically-driven  tuning  fork  which  serves  to  make 
and  break  contact  with  the  points  F  and  G  in  rapid  alternation. 
When  the  tuning  fork  makes  contact  at  F  the  polarizing  current 


B— 


Fig.  100. 


Fig.  101. 


flows  through  A,  polarizing  the  electrodes;  when  contact  is  made 
at  Gj  the  counter  electromotive  force  due  to  this  polarization 
causes  a  current  to  flow  through  D  and  C.  The  counter  electro- 
motive force  is  balanced  by  varying  D,  until  C  indicates  zero 
current:  the  potential  of  D  is  then  equal  to  the  electromotive  force 
of  polarization.  Just  as  the  electromotive]  force  of  a  galvanic  cell 
is  due  to  the  combined  action  of  several  differences  of  potential, 
so  also  the  electromotive  force  of  polarization  is  due  to  the  individ- 
ual differences  of  potential  located  at  the  electrodes.  The  method 
employed  for  the  measurement  of  polarization  at  a  single  elec- 
trode was  devised  by  Fuchs,  and  is  illustrated  diagrammatically 
in  Fig.  101.  Into  the  vessel  containing  the  electrolyte,  dip  the 
*  Zeit.  phys.  Chem.,  5,  469  (1890). 


434  THEORETICAL  CHEMISTRY 

two  electrodes  A  and  B,  and  the  side  tube  of  the  calomel  normal 
electrode  C.  D  is  the  source  of  external  electromotive  force,  E  is 
a  capillary  electrometer,  and  F  is  a  source  of  variable  potential. 
Before  closing  the  external  circuit  DAB,  the  potential  of  the 
electrode  B  against  the  solution  is  first  measured.  Then  the  cir- 
cuit DAB  is  closed,  thereby  polarizing  the  electrode  B.  The 
external  circuit  is  again  broken  and  the  potential  of  B  against  the 
solution  remeasured.  The  difference  between  the  final  and  initial 
values  of  the  electrode  potential  gives  the  polarization  at  B.  In 
like  manner,  the  polarization  at  the  electrode  A  can  be  measured. 
The  small  amount  of  electricity  which  is  necessary  to  polarize 
an  electrode  is  termed  the  polarization  capacity  of  the  electrode. 
This  factor  is  dependent  upon  the  extent  of  surface  of  the  electrode, 
and  also  upon  the  nature  of  the  metal  of  which  the  electrode  is 
made.  For  electrodes  of  equal  surface,  the  polarization  capacity 
of  palladium  is  greater  than  that  of  platinum,  when  hydrogen 
is  [liberated  on  each.  The  solubility  of  hydrogen  is  greater  in 
palladium  than  in  platinum,  and  consequently,  because  a  larger 
amount  of  hydrogen  is  dissolved,  a  greater  quantity  of  electricity 
will  be  required  to  bring  the  pressure  of  the  hydrogen  up  to  that 
of  the  hydrogen  dissolved  in  the  platinum.  If,  through  the 
processes  of  solution  or  diffusion,  or  through  chemical  action,  the 
substance  which  causes  the  polarization  is  removed,  the  electrode 
is  said  to  be  depolarized.  Thus,  when  a  reducing  agent,  such  as 
ferrous  chloride,  is  electrolyzed,  the  oxygen  liberated  at  the  anode 
immediately  combines  with  the  electrolyte,  forming  ferric  chloride 
and  preventing  polarization  of  the  electrode. 

If  water  be  electrolyzed  between  platinum  electrodes,  the  cathode 
becomes  saturated  with  hydrogen  and  the  anode  with  oxygen, 
until  when  the  electromotive  force  of  polarization  becomes  equal 
to  that  of  the  external  circuit,  the  current  ceases.  The  two 
gases,  hydrogen  and  oxygen,  are  soluble,  however,  and  conse- 
quently diffuse  away  from  the  electrodes,  either  escaping  from 
the  solution  or  recombining  to  form  water.  In  order  to  compen- 
sate for  this  continuous  loss  of  gas  at  the  electrodes,  a  small  current 
continues  to  flow,  thus  maintaining  the  initial  electromotive  force 
constant.  This  small  current  is  termed  the  residual  current. 


ELECTROLYSIS  AND  POLARIZATION 


435 


If  oxygen  is  bubbled  over  the  surface  of  the  cathode  during  elec- 
trolysis, the  hydrogen  is  removed  as  rapidly  as  it  is  liberated. 
Such  an  electrode  on  which  no  new  substance  is  formed  during 
electrolysis  is  called  an  unpolarizable  electrode. 

Decomposition  Potentials.  The  decomposition  potential  of  an 
electrolyte  can  be  determined,  as  has  already  been  pointed  out, 
by  immersing  two  platinum  electrodes  in  the  solution  and  connect- 
ing with  a  source  of  electricity,  the  electromotive  force  of  which 
can  be  varied  at  will.  The  voltage  is  gradually  increased  and  the 
corresponding  current  is  observed.  The  current  increases  at  first 


Electromotive  Force 
Fig.  102. 


and  then  drops  almost  to  zero  every  time  the  voltage  is  raised, 
until  the  decomposition  potential  is  reached.  Beyond  this  point 
the  current  is  directly  proportional  to  the  electromotive  force. 
If  the  applied  electromotive  forces  are  plotted  as  abscissae  and 
the  corresponding  currents  as  ordinates,  we  obtain  curves  of  the 
form  shown  in  Fig.  102.  Some  of  the  decomposition  potentials 
of  molar  solutions  determined  by  Le  Blanc  *  are  given  in  the  accom- 
panying tables. 

*  Zeit.  phys.  Chem.,  8,  299  (1891). 


436 


THEORETICAL  CHEMISTRY 
SALTS. 


Salt. 

Decomp. 
Potential. 

Salt. 

Decomp. 
Potential. 

ZnSO4 

Volts. 

2  35 

Cd(N03)2 

Volts. 

1  98 

ZnBr2  

1  80 

CdSO4                 .     .    . 

2.03 

NiSO4  

2.09 

CdCl2  

1.88 

NiCl2  

1.85 

CoSO4  

1.92 

Pb(NO3)2.. 

1.52 

CoCl2  

1.78 

AgN03  

0.70 

ACIDS. 


Acid. 

Decomp. 
Potential. 

H2SO4.  . 

Volts. 
1.67 

HNO3.. 

1.69 

H3PO4 

1  70 

CH2C1.COOH.. 

1.72 

CHC12.COOH  
CH2(COOH)2  
HC1O4 

1.66 
1.69 
1  65 

HC1 

1  31 

(COOH)2..   . 

0  95 

HBr  

0.94 

HI  

0.52 

BASES. 


Base. 

Decomp. 
Potential. 

NaOH 

Volts. 
1   69 

KOH 

1  67 

NH4OH  

1.74 

It  will  be  observed  that  while  there  is  considerable  variation  in 
the  decomposition  potentials  of  salts,  there  is  very  little  variation 
in  the  decomposition  potentials  of  acids  and  bases.  There  is  a 
maximum  value  of  about  1.70  volts  to  which  many  acids  and 
bases  closely  approximate.  It  is  found  that  all  acids  and  bases 
which  decompose  at  1.70  volts  give  off  hydrogen  and  oxygen  at 


ELECTROLYSIS  AND  POLARIZATION 


437 


the  electrodes.  Those  acids  and  bases  which  decompose  at 
potentials  less  than  the  maximum,  do  not  liberate  hydrogen  and 
oxygen.  When  their  solutions  are  sufficiently  diluted,  however, 
hydrogen  and  oxygen  are  evolved  and  the  decomposition  potential 
rises  to  the  maximum  value.  Thus,  Le  Blanc  found  the  following 
values  for  the  decomposition  potential  of  different  dilutions  of 
hydrochloric  acid. 


Concentration. 

Decomp. 
Potential. 

Volts. 

2mHCl 

1.26 

ImHCl 

1.34 

ImHCl 

1.41 

rkmHCl 

1.62 

^rnHCl 

1.69 

When  2  m  hydrochloric  acid  is  electrolyzed,  hydrogen  and 
chlorine  are  given  off  at  the  electrodes,  whereas  when  the  concen- 
tration of  the  acid  is  reduced  to  1/32  m,  hydrogen  and  oxygen 
are  the  products  of  electrolysis,  and  the  decomposition  potential 
increases  to  1.70  volts.  It  is  found  that  the  values  of  the  decom- 
position potentials  vary  slightly  with  the  nature  of  the  electrodes 
used.  The  above  values  were  determined  with  platinum  elec- 
trodes. 

The  Theory  of  Polarization.  Our  knowledge  of  the  process 
taking  place  at  the  electrodes  during  electrolysis  is  largely  due 
to  the  investigations  of  Le  Blanc.  He  determined  the  electro- 
motive force  of  polarization  at  each  electrode,  varying  the  external 
electromotive  force  from  zero  up  to  the  decomposition  potential 
of  the  solution.  When  the  decomposition  value  was  reached,  he 
found  the  potential  of  the  electrode  against  the  solution  to  be  the 
same  as  the  difference  of  potential  between  the  solution  and  the 
element  liberated  at  the  electrode.  Thus,  the  decomposition 
potential  of  a  molar  solution  of  zinc  sulphate  is  2.35  volts;  the 
corresponding  difference  of  potential  between  the  electrode  and 
the  solution  is  found  to  be  0.493  volt.  If  a  piece  of  pure  zinc 
is  immersed  in  a  molar  solution  of  zinc  sulphate,  the  difference 


438  THEORETICAL  CHEMISTRY 

of  potential  is  found  to  be  0.493  volt,  the  metal  being  negative 
to  the  solution.  It  frequently  happens  that  the  electrode  exhibits 
the  potential  due  to  the  deposited  metal  before  the  decomposition 
point  of  the  solution  is  reached.  For  example,  in  a  molar  solution 
of  silver  nitrate  the  electrode  acquires  the  potential  of  pure  silver 
in  molar  silver  nitrate  before  the  decomposition  value,  0.70  volt, 
is  reached.  This  is  due  to  the  negative  solution  pressure  of  the 
silver  which  causes  the  deposition  of  the  ions  of  the  metal  without 
the  application  of  any  external  electromotive  force.  When  an 
indifferent  electrode,  such  as  platinum,  is  immersed  in  a  solution 
of  a  salt,  a  very  small  amount  of  ionic  deposition  must  occur, 
otherwise,  according  to  the  Nernst  equation,  an  infinite  electro- 
motive force  must  be  established.  Thus,  in  the  equation 

RT 


if  the  solution  pressure  P  =  0,  it  is  evident  that  TT  =  oo  and  a 
perpetual  motion  must  result.  We  are  thus  forced  to  the  con- 
clusion that  when  an  indifferent  electrode  is  immersed  in  a  salt 
solution,  ions  will  continue  to  separate  upon  it  until  the  tendency 
for  the  deposited  metal  to  go  back  into  solution  in  the  ionic  state 
exactly  counterbalances  the  tendency  to  separation.  Hence,  the 
electrode  will  become  positive  toward  the  solution.  The  magnitude 
of  this  difference  of  potential  will  be  dependent  upon  the  amount 
of  metal  deposited.  It  is  to  be  noted  that  this  difference  of 
potential  need  not  be  equal  to  that  between  the  massive  metal 
and  the  solution.  If  the  electrodes  be  connected  with  an  external 
source  of  electromotive  force,  the  value  of  which  can  be  varied  at 
will,  and  a  small  electromotive  force  be  applied,  more  metal  will 
separate  on  the  cathode.  This  will  cause  an  increase  in  the  solu- 
tion pressure  P,  tending  to  offset  further  deposition.  A  still 
further  increase  in  the  external  electromotive  force  will  cause  the 
deposition  of  more  metal,  and  as  a  result  of  the  corresponding 
increase  in  P,  further  deposition  at  that  voltage  will  be  pre- 
vented. Ultimately,  when  the  applied  electromotive  force  is 
such  that  P  acquires  its  maximum  value,  equivalent  to  that  of 
the  massive  metal,  continuous  deposition  will  occur.  An  exactly 


ELECTROLYSIS  AND  POLARIZATION  439 

analogous  process  takes  place  at  the  anode.  If  a  gas  is  liberated, 
its  concentration  steadily  increases  until  the  maximum  pressure 
is  reached,  when  it  will  escape  from  the  solution.  When  strong 
currents  are  employed,  P  does  not  remain  constant,  as  has  been 
assumed  above,  but  gradually  diminishes  causing  the  difference  of 
potential  at  the  electrode  to  increase. 

From  the  above  considerations  it  becomes  clear  why  a  definite 
electromotive  force  is  necessary  to  bring  about  a  continuous 
decomposition  of  an  electrolyte:  this  will  only  take  place  when 
the  concentrations  of  the  substances  separating  at  the  electrodes 
have  attained  their  maximum  values.  When  the  decomposition 
point  is  reached,  the  electrode  exhibits  the  potential  characteristic 
of  the  massive  metal.  It  is  evident  from  the  behavior  of  silver 
nitrate  and  the  salts  of  other  metals  having  negative  solution 
pressures,  that  the  maximum  values  of  concentration  at  the 
the  electrodes  need  not  necessarily  be  attained  simultaneously. 

When  the  products  of  electrolysis  are  gaseous,  the  value  of  the 
decomposition  potential  depends  upon  the  nature  of  the  electrodes. 
Thus,  the  cell 

PtH,  -  m  H2S04  -  Pto2 

gives  an  electromotive  force  of  1.07  volts  if  platinized  platinum 
electrodes  are  used.  If  an  external  electromotive  force  slightly 
greater  than  1 .07  volts  be  applied  to  this  cell  in  the  reverse  direc- 
tion, water  will  be  steadily  decomposed,  hydrogen  and  oxygen 
being  evolved  at  the  electrodes.  If  on  the  other  hand,  the  plati- 
nized electrodes  are  replaced  by  electrodes  of  polished  platinum, 
the  decomposition  potential  %rises  to  1.68  volts.  The  reverse 
electromotive  force  of  polarization,  however,  is  only  1.07  volts. 
That  is,  the  liberation  of  gas  at  a  polished  platinum  electrode  is 
an  irreversible  process.  The  difference  in  the  behavior  of  the 
two  electrodes  is  explicable  when,  it  is  remembered  that  platinum 
is  capable  of  occluding  large  amounts  of  gas.  A  platinized  elec- 
trode absorbs  the  liberated  gas  very  slowly,  and  when  thoroughly 
saturated,  if  it  is  not  entirely  immersed  in  the  solution,  it  gradually 
gives  up  the  gas  by  diffusion,  no  bubbles  being  formed.  Thus,  if 
the  external  electromotive  force  be  raised  to  1.07  volts,  the  system 


440  THEORETICAL  CHEMISTRY 

will  be  in  equilibrium,  while  if  the  applied  electromotive  force  be 
greater  or  less  than  the  equilibrium  value,  a  current  will  flow  in 
one  direction  or  the  other,  gas  being  either  liberated  or  dissolved. 
In  other  words,  the  cell  is  completely  reversible. 

Where  polished  platinum  or  gold  electrodes  are  used,  however, 
the  decomposition  potential  is,  as  has  been  stated,  1.68  volts. 
Polished  electrodes  have  relatively  small  absorbing  power.  Hence, 
if  an  electromotive  force  between  1.07  and  1.68  volts  be  applied, 
the  gases  cannot  diffuse  away  from  the  electrode  rapidly  enough, 
and,  when  the  solution  in  the  vicinity  of  the  electrodes  becomes 
saturated  with  gas,  the  current  ceases  to  flow. 

A  very  slow  process  of  diffusion  from  the  solution  into  the  air 
is  constantly  taking  place  however,  and  this  permits  the  continu- 
ous evolution  of  an  exceedingly  small  amount  of  gas,  while  a  corre- 
spondingly small  current  traverses  the  solution. 

In  order  to  produce  a  steady  electrolysis,  it  is  necessary  to  raise 
the  external  electromotive  force  to  such  an  intensity  that  it  is 
able  to  bring  about  the  formation  of  bubbles  at  the  surface  of  the 
electrodes.  This  calls  for  the  expenditure  of  an  amount  of  work 
depending  upon  the  condition  of  the  electrode  surfaces,  the  sur- 
face tension  of  the  solution,  and  various  other  factors.  In  cases 
where  bubbles  are  formed,  a  portion  of  the  available  energy  of 
the  chemical  process  is  not  expended  in  effecting  electrical  separa- 
tion; consequently  the  reverse  electromotive  force  is  less  than  the 
applied,  and  the  system  is  irreversible. 

The  reactions  at  the  electrodes  are  catalytically  accelerated  by 
the  metal  of  which  the  electrodes  are  made.  Thus,  platinized 
platinum  is  the  most  effective  catalyst  for  the  reaction  represented 
by  the  equation 


Hydrogen  is  liberated  on  platinized  platinum  at  the  potential 
0  volt,  on  polished  platinum  at  0.09  volt,  and  on  zinc  at  0.70  volt. 
The  electromotive  force  necessary  to  overcome  the  resistance  of 
the  chemical  reaction  at  an  electrode  is  termed  the  overvoltage. 
Thus,  we  say  that  hydrogen  is  liberated  on  polished  platinum  with 
an  overvoltage  of  0.09  volt,  and  on  zinc  with  an  overvoltage  of 


ELECTROLYSIS  AND   POLARIZATION 


441 


0.70  volt.  The  following  table  gives  the  overvoltage  necessary 
for  the  liberation  of  hydrogen  and  oxygen  on  electrodes  of  differ- 
ent metals. 

ELECTRODE  OVERVOLTAGES. 


Hydrogen  Liberation. 

Oxygen  Liberation. 

Metal. 

Overvoltage. 

Metal. 

Overvoltage. 

Pt  (platinized) 

0.00 
0.01 
0.08 
0.09 
0.15 
0.21 
0.23 
0.46 
0.53 
0.64 
0.70 
0.78 

Au  

.75 

.67 
.65 
.65 
.63 
1.53 
1.48 
1.47 
1.47 
1.36 
1.35 
1.28 

Au 

Pt  (polished)  
Pd.  . 

Fe  (in  NaOH)     . 

Pt  (polished) 

Cd 

Aff 

Ag 

Ni 

Pb 

Cu 

Cu 

Pd 

Fe 

Sn 

Pt  (platinized)  

Pb                  

Co  

Zn               

Ni  (polished)  

He 

Ni  (spongy)  

Primary  Decomposition  of  Water  in  Electrolysis.  The  decom- 
position potential  of  an  electrolyte  giving  off  hydrogen  and  oxygen 
at  the  electrodes,  is  dependent  upon  the  concentrations  of  the  two 
ions,  H*  and  OH',  and  is  independent  of  the  nature  of  the  electrolyte. 
As  has  already  been  stated,  the  decomposition  potential  of  all 
acids  and  bases  giving  off  hydrogen  and  oxygen  approximates  to 
1.70  volts.  According  to  the  law  of  mass  action,  the  product  of 
the  concentrations  of  the  H"  and  OH'  ions  is  constant  and  inde- 
pendent of  the  other  substances  which  may  be  present;  hence, 
although  the  potentials  of  the  individual  electrodes  may  differ 
considerably,  their  sum  remains  practically  constant. 

Excluding  solutions  of  salts  which  undergo  reduction  by  hydro- 
gen, and  solutions  of  chlorides,  bromides,  and  iodides  reducible 
by  oxygen,  the  ions  H'  and  OH',  according  to  Le  Blanc,  are  to 
be  regarded  as  the  sole  factors  in  the  electrolysis  of  solutions,  and 
not  the  ions  of  the  dissolved  electrolyte.  In  other  words,  elec- 
trolysis involves  a  primary  decomposition  of  water. 

The  electrical  conductance  of  the  solution  is  due  to  the  ions  of 
the  electrolyte  together  with  the  ions  of  water,  but  at  the  electrode 


442  THEORETICAL  CHEMISTRY 

that  process  takes  place  which  involves  the  expenditure  of  the 
minimum  amount  of  energy,  and  this  is,  under  ordinary  conditions 
the  separation  of  the  H*  and  OH'  ions. 

Thus,  when  a  solution  of  potassium  sulphate  is  electrolyzed, 
only  a  moderately  strong  current  being  used,  it  is  not  rational  to 
assume  the  discharge  of  the  K*  and  SO/'  ions  at  the  electrodes, 
and  then  subsequent  reaction  between  these  discharged  ions  and 
water.  This  may  be  made  clear  by  considering  the  process  taking 
place  at  the  cathode.  According  to  the  explanation  based  upon 
so-called  "secondary  action,"  the  K*  ions  give  up  their  positive 
charges  to  the  electrode  and  then  react  with  water  as  indicated 
by  the  equation 

K  +  H*  +  OH'  ->K"  +  OH'  +  H. 

This  explanation  involves  the  transfer  to  the  potassium  atom  of 
the  positive  charge  of  the  H*  ion  of  water;  this  can  only  take  place 
if  the  H"  ion  holds  its  charge  less  tenaciously  than  the  K"  ion. 
Hence,  if  the  H"  ion  parts  with  its  charge  more  readily  than  the 
K*  ion,  the  former  will  be  discharged  primarily  at  the  cathode. 
Similar  reasoning  may  be  employed  to  explain  the  action  at  the 
anode.  Therefore,  in  electrolysis  all  of  the  ions  participate  in 
conducting  the  current  and  collect  around  the  electrodes,  but 
since  the  H*  and  OH'  ions  separate  more  easily,  these  are  dis- 
charged. With  stronger  currents  it  is  possible  to  cause  the  separ- 
ation of  the  K*  and  SO/'  ions  also,  since  the  number  of  H"  and  OH' 
ions  present  is  too  small  to  carry  all  of  the  current,  and  the  energy 
required  to  discharge  the  ions  of  the  electrolyte  is  less  than  that 
necessary  to  remove  the  small  number  of  residual  H*  and  OH' 
ions.  The  formation  and  decomposition  of  water  are  reversible 
processes,  so  that  no  loss  of  energy  is  involved,  as  would  be  the 
case  if  secondary  actions  occurred. 

Electrolytic  Separation  of  the  Metals.  Freudenberg  *  was  the 
first  to  recognize  the  possibility  of  effecting  the  quantitative 
separation  of  different  metals  by  means  of  graded  electromotive 
forces.  He  showed  that  it  was  only  necessary  to  select  a  salt  of 
each  metal  the  decomposition  potentials  of  which  differ  as  widely 
as  possible,  and  electrolyze  at  an  electromotive  force  intermediate 
*  Zeit.  phys.  Chem.,  12,  97  (1893). 


ELECTROLYSIS  AND  POLARIZATION 


443 


between  these  potentials.  The  salt  having  the  lower  decomposi- 
tion potential  will  decompose  first,  and  when  the  deposition  of 
the  metal  is  complete,  the  current  will  practically  cease;  then  if 
the  applied  electromotive  force  be  raised  above  the  decomposition 
potential  of  the  second  salt,  the  second  metal  will  be  deposited. 
In  practice  it  is  found  necessary  to  increase  the  applied  electro- 
motive force  slightly  because  of  the  gradual  decrease  in  the  num- 
ber of  ions  of  the  salt  having  the  lower  decomposition  potential. 
The  amount  of  this  increase  may  be  readily  calculated  from  the 
familiar  equation 

RT,       P 


Suppose  a  mixture  of  the  nitrates  of  cadmium,  lead  and  silver  is 
subjected  to  electrolysis,  the  decomposition  potentials  of  the  salts 
being  as  follows:  —  Cd(N03)2  =  1.98  volts,  Pb(N03)2  =  1.52 
volts,  and  AgN03  =  0.70  volt.  The  applied  electromotive  force 
is  made  a  little  less  than  1  volt  and  all  of  the  silver  is  deposited; 
then  the  electromotive  force  is  raised  to  about  1.6  volts,  thus 
depositing  all  of  the  lead;  and  finally,  with  an  electromotive  force 
of  about  2  volts  the  cadmium  is  deposited. 

In  the  subjoined  table  are  given  the  separation  potentials  of 
some  of  the  ions,  the  separation  potential  of  the  H'  ion  being 
assumed  to  be  equal  to  zero. 

SEPARATION  VALUES  OF  IONS  FOR  MOLAR  CONCENTRA- 

TION. 


Ion. 

Separation 
Potential. 

Ion. 

Separation 
Potential. 

AgV. 

—0  78 

I' 

0  52 

Cu".. 

-0.34 

Br' 

0  94 

H*.. 

0.0 

O" 

1  08  (in  acid) 

Pb".. 

0.17 

01' 

1  31 

Cd".. 

0.38 

OH' 

1  68  (in  acid) 

Zn". 

0  74 

OH' 

0  88  (in  base) 

SO4".  . 

1  9 

According  to  this  table  the  decomposition  potential  of  water  is 
equal  to  the  sum  of  the  separation  potentials  of  its  ions,  or  1.68 
volts. 


CHAPTER  XIX. 
ACTINOCHEMISTRY. 

Radiant  Energy.  The  visible  portion  of  the  spectrum  is  com- 
prised between  the  extreme  red  at  one  end  and  the  extreme  violet 
at  the  other;  the  wave-length  corresponding  to  the  former  is 
approximately  0.7  micron,  while  that  corresponding  to  the  latter 
is  about  0.4  micron.  The  visible  portion  of  the  spectrum,  how- 
ever, is  but  a  small  fraction  of  the  entire  spectrum.  Beyond  the 
red  of  the  visible  spectrum  lies  the  region  of  the  so-called  infra- 
red, comprising  all  wave-lengths  from  0.76  micron  up  to  300  mi- 
crons. Beyond  the  infra-red,  between  300  and  2000  microns  is 
an  unmeasured  region,  which  is  succeeded  by  the  region  of  elec- 
trical waves,  extending  from  2000  microns  to  an  undetermined 
maximum.  On  the  other  hand,  extending  beyond  the  violet  of 
the  visible  spectrum  is  the  so-called  ultra-violet  or  actinic  region, 
comprising  all  wave-lengths,  between  0.4  micron  and  0.1  micron. 
It  thus  appears  that  heat,  light  and  electricity  are  all  forms  of 
radiant  energy,  the  only  distinction  between  them  being  a  differ- 
ence in  wave-length.  Very  little  is  known  concerning  radiant 
energy,  and  up  to  the  present  time  all  attempts  to  resolve  it  into 
a  capacity  and  an  intensity  factor  have  failed. 

Whatever  may  be  the  nature  of  this  form  of  energy,  we  know 
that  the  effects  produced  by  it  are  dependent  upon  the  wave- 
length. We  have  already  devoted  several  chapters  to  the  con- 
sideration of  thermochemistry  and  electrochemistry,  and  it  now 
remains  to  study  very  briefly  the  connection  between  chemical 
energy  and  that  subdivision  of  radiant  energy  called  light.  This 
branch  of  theoretical  chemistry  is  termed  photochemistry,  (<£oos  = 
light),  or  actinochemistry  (owns  —  a  ray),  the  latter  term  being 
preferable.  The  ultra-violet  or  actinic  rays  are  the  most  active 
chemically,  although  light  of  every  wave-length  including  the 
invisible  infra-red  is  capable  of  producing  chemical  action.  When 
light  falls  upon  a  substance,  a  portion  is  reflected,  a  portion  is 

444 


ACTINOCHEMISTRY  445 

absorbed,  and  a  portion  is  transmitted.  It  has  been  shown  that 
only  that  portion  of  the  incident  radiant  energy  which  is  absorbed 
is  effective  in  producing  chemical  change. 

Radiant  energy  has  been  shown  by  Lebedew,*  and  Nichols  and 
Hull  f  to  exert  a  definite,  though  extremely  small  pressure. 
Thus,  the  pressure  of  solar  radiations  on  the  earth  is  equivalent 
to  that  of  a  column  of  mercury  1.4  X  10~9  mm.  high. 

The  amount  of  radiant  energy  received  by  the  earth  from  the 
sun  has  been  estimated  by  Le  Chatelier  to  be  2,100,000  calories 
per  square  meter  per  annum.  This  represents  only  a  very  small 
fraction  of  the  total  energy  radiated  by  the  sun.  If  the  radiant 
energy  received  by  the  earth  could  be  transformed  without  loss 
into  mechanical  energy,  it  would  be  equivalent  to  approximately 
0.3  horse-power  per  square  meter.  Notwithstanding  the  many 
attempts  which  have  been  made  to  harness  solar  energy,  no  prac- 
tical results  have  thus  far  been  achieved,  and  it  is  an  easy  matter 
to  demonstrate  that  there  is  slight  chance  of  obtaining  an  efficiency 
greater  than  one  per  cent  of  the  radiant  energy  received. 

In  a  recent  address,  Professor  Ciamician  {  made  an  interesting 
comparison  of  the  amount  of  energy  stored  up  in  fossil  fuel  with 
the  actual  energy  received  from  the  sun;  he  said,  "  Assuming  that 
the  solar  constant  (the  number  of  calories  received  per  square 
centimeter  per  minute)  is  three  small  calories  a  minute  per  square 
centimeter,  that  is,  thirty  large  calories  a  minute  per  square  meter 
or  about  1800  large  calories  an  hour,  we  may  compare  this  quantity 
of  heat  with  that  produced  by  the  complete  combustion  of  a  kilo- 
gram of  coal,  which  is  8000  calories.  Assuming  for  the  tropics  a 
day  of  only  six  hours  sunshine  we  should  have,  for  the  day,  an 
amount  of  heat  equivalent  to  that  furnished  by  1.35  kilograms  of 
coal,  or  1  kilogram  in  round  numbers.  For  a  square  kilometer, 
we  should  have  a  quantity  of  heat  equivalent  to  that  produced 
by  the  combustion  of  1000  tons  of  coal.  A  surface  of  only  10,000 
square  kilometers  receives  in  a  year,  calculating  a  day  of  only  six 

*  Rapp.  pres  au  Congres  de  Physique,  2,  133  (1900^ 
f  Phys.  Rev.,  13,  293  (1901). 

t  Lecture  before  Eighth  International  Congress  of  Applied  Chemistry, 
New  York,  1912. 


446  THEORETICAL  CHEMISTRY 

hours,  a  quantity  of  heat  that  corresponds  to  that  produced  by 
the  burning  of  3650  million  tons  of  coal,  in  round  numbers  three 
billion  tons.  The  quantity  of  coal  produced  annually  (1909)  in 
the  mines  of  Europe  and  America  is  calculated  at  about  925  million 
tons,  and  adding  to  this  175  million  tons  of  lignite,  we  reach  1100 
million  tons,  or  a  little  over  one  billion.  Even  making  allowances 
for  the  absorption  of  heat  on  the  part  of  the  atmosphere  and  for 
other  circumstances,  we  see  that  the  solar  energy  that  reaches  a 
small  tropical  country  —  say  of  the  size  of  Latium  —  is  equal 
annually  to  the  energy  produced  by  the  entire  amount  of  coal 
mined  in  the  world.  The  desert  of  Sahara  with  its  six  million 
square  kilometers  receives  daily,  solar  energy  equivalent  to  six 
billion  tons  of  coal.  This  enormous  quantity  of  energy  that  the 
earth  receives  from  the  sun,  in  comparison  with  which  the  part 
which  has  been  .stored  up  by  the  plants  in  the  geological  periods 
is  almost  negligible,  is  largely  wasted."  It  is  evident  that  if  only 
a  small  fraction  of  this  lost  energy  could  be  transformed  into  work 
an  inestimable  blessing  would  be  conferred  upon  mankind. 

Actinometers.  A  number  of  different  forms  of  apparatus  have 
been  devised  for  measuring  the  chemical  action  of  light:  such 
instruments  are  known  as  actinometers.  It  should  be  borne  in 
mind,  however,  that  the  results  of  actinometric  measurements  are 
only  relative. 

The  Hydrogen-Chlorine  Actinometer.  This  instrument  is  based 
upon  the  well-known  fact  that  the  speed  of  the  reaction  between 
hydrogen  and  chlorine  varies  greatly  with  the  intensity  of  illu- 
mination. Bunsen  and  Roscoe,*  guided  by  the  experiments  of 
Draper,  constructed  an  actinometer  in  which  the  rate  of  combina- 
tion of  hydrogen  and  chlorine  could  be  measured  by  allowing 
the  hydrochloric  acid  formed  to  dissolve  in  water,  and  noting  the 
D  „  xrx  A 


Fig.  103. 

resulting  diminution  of  volume.     A  diagram  of  this  apparatus 

is  given  in  Fig.  103.     The  apparatus  is  filled  with  a  mixture  of 

*  Pogg.  Ann.,  100,  43  (1857);  101,  235  (1857). 


ACTINOCHEMISTRY  447 

equal  parts  of  hydrogen  and  chlorine,  obtained  by  the  electrolysis 
of  a  solution  of  hydrochloric  acid.  The  bulb  A,  containing  water, 
is  connected  at  one  end  with  a  tube  fitted  with  a  stop-cock  B,  and 
at  the  other  end  with  a  horizontal  tube  terminating  in  a  reservoir 
D,  which  also  contains  water.  When  the  water  has  become  sat- 
urated with  the  constituents  of  the  gaseous  mixture,  B  is  closed 
and  the  entire  apparatus  is  protected  from  light.  When  it  is  desired 
to  measure  the  photochemical  action  of  a  source  of  light,  the  bulb 
A  is  uncovered  and  the  light  is  allowed  to  fall  upon  it.  Some  of 
the  hydrogen  and  chlorine  will  combine,  and  the  hydrochloric 
acid  formed  will  be  absorbed  by  the  water  in  A;  the  column  of 
water  in  the  horizontal  tube  will  move  to  the  right,  the  magnitude 
of  the  movement  being  measured  on  the  scale  C.  In  this  way  the 
amount  of  photochemical  action  can  be  determined.  An  objec- 
tion to  the  use  of  hydrogen  and  chlorine  in  the  actinometer,  is  the 
danger  of  violent  explosions  when  the  illumination  is  too  intense. 
To  remove  this  objection,  Burnett*  replaced  the  hydrogen  of  the 
mixture  by  carbon  monoxide. 

The  Silver  Chloride  Actinometer.  This  consists  of  a  strip  of 
paper  coated  with  silver  chloride;  a  portion  of  it  is  exposed  to 
the  light,  and  the  time  required  for.  it  to  darken  to  a  definite  shade 
is  noted.  The  intensity  of  the  light  varies  inversely  as  the  time 
of  exposure.  This  is  a  somewhat  crude  device  and  the  results 
obtained  are  not  very  reliable,  although  Bunsen  and  Roscoe  f 
used  it  in  some  of  their  investigations. 

The  Mercuric  Oxalate  Actinometer.  The  reaction  between 
mercuric  chloride  and  ammonium  oxalate  has  been  employed  to 
measure  the  intensity  of  light.  These  substances  react  in  the 
following  manner  when  exposed  to  light :  — 

2  HgCl2  +  (NH4)2C204  =  2  HgCl  +  2  C02  +  2  NH4C1. 

The  extent  to  which  the  reaction  proceeds  can  be  determined  by 
measuring  the  amount  of  mercurous  chloride  precipitated,  or  the 
volume  of  carbon  dioxide  formed.  Either  of  these  quantities  is 
proportional  to  the  intensity  of  the  incident  light. 

*  Phil.  Mag.  [4],  20,  406  (1860). 

t  Pogg.  Ann.,  117,  529  (1862);  124,  353  (1865);  132,  404  (1867). 


448  THEORETICAL  CHEMISTRY 

Electrical  Actinometers.  Becquerel  *  showed  that  if  two  silver 
electrodes  coated  with  silver  iodide  are  immersed  in  dilute  sulphuric 
acid,  and  one  electrode  is  illuminated  while  the  other  is  screened 
from  the  light,  an  electric  current  passes.  The  deflection  of  the 
needle  of  a  galvanometer  included  in  the  circuit,  may  be  taken  as 
a  measure  of  the  intensity  of  the  incident  light.  Rigollot  f  has 
constructed  an  electrical  actinometer  in  which  the  silver  plates 
used  by  Becquerel  are  replaced  by  two  oxidized  copper  electrodes, 
while  instead  of  dilute  sulphuric  acid  a  dilute  solution  of  sodium 
chloride  is  used.  The  action  of  the  Becquerel  actinometer  has 
been  explained  by  Ostwald  as  follows :  —  The  incident  light 
lessens  the  stability  of  the  silver  iodide  which  undergoes  ioniza- 
tion  according  to  the  equation 

Agl-^Ag+T: 

the  Ag"  ions  give  up  their  charges  to  the  electrode  while  the  V 
ions  enter  the  solution.  For  every  Ag*  ion  which  is  discharged  at 
the  illuminated  electrode,  an  equal  number  of  Ag*  ions  enter  the 
solution  at  the  darkened  electrode,  thus  charging  the  latter 
negatively.  Hence  the  current  flows  in  the  solution  from  the 
darkened  to  the  illuminated  electrode. 

Photochemical  Absorption.  The  greater  part  of  the  radiant 
energy  absorbed  by  a  substance  is  transformed  into  heat,  while  a 
small  fraction  may  become  effective  in  bringing  about 'chemical 
changes.  Such  changes  are  said  to  be  due  to  photochemical  absorp- 
tion. The  amount  of  photochemical  absorption  varies  with  the 
nature  of  the  substance  and  also  with  the  wave-length  of  the 
incident  light.  The  bleaching  of  colored  fabrics  by  sunlight,  and 
the  effects  of  ultra-violet  light  on  the  skin,  may  be  mentioned 
as  examples  of  photochemical  absorption.!  The  formation  of 
chlorophyll,  and  other  substances  synthesized  by  growing  plants, 
is  due  primarily  to  photochemical  absorption. 

*  Ann.  Chim.  Phys.  [3],  9,  257  (1843);  32,  176  (1881). 
t  Jour,  de  Phys.  [3],  6,  520  (1897). 

J  In  this  connection  the  student  will  find  Finsen's  "Phototherapy,"  inter- 
esting reading. 


ACTINOCHEMISTRY  449 

Photochemical  Induction.  Becquerel  *  discovered  that  silver 
chloride  which  had  been  precipitated  in  the  dark  was  only  sensitive 
to  short  wave-lengths  of  light,  whereas  silver  chloride  which  had 
been  exposed  for  a  few  moments  to  sun-light,  became  sensitive 
to  all  wave-lengths  in  the  visible  spectrum  and  to  the  shorter 
wave-lengths  of  the  infra-red.  This  phenomenon,  which  has  been 
observed  with  various  substances,  was  first  studied  systematically 
by  Bunsen  and  Roscoef  who  termed  it  photochemical  induction. 
Thus,  employing  the  hydrogen-chlorine  actinometer  containing 
freshly  prepared  gases,  they  found  that  under  constant  illumi- 
nation, the  velocity  of  formation  of  hydrochloric  acid  which  was 
hardly  appreciable  at  first,  increased  rapidly  to  a  maximum  and 
then  remained  constant.  The  interval  of  time  required  for  the 
reaction  to  attain  its  maximum  velocity  is  known  as  the  period  of 
induction. 

In  one  experiment  with  the  hydrogen-chlorine  actinometer 
using  the  light  from  the  zenith  of  a  cloudless  sky,  Bunsen  and 
Roscoe  found  the  period  of  induction  to  be  approximately  11 
minutes.  If,  after  the  maximum  velocity  of  reaction  had  been 
reached,  the  gaseous  mixture  was  placed  in  the  dark,  it  was  found 
that  after  30  minutes,  the  gases  remaining  uncombined  had  returned 
to  their  initial  condition  and  approximately  the  same  period  of 
induction  was  observed  on  re-exposure.  Bunsen  and  Roscoe  also 
showed  that  if  the  hydrogen  and  chlorine  were  each  exposed  sepa- 
rately, and  then  mixed,  the  period  of  induction  remained  unchanged. 
That  is,  the  incident  light  is  apparently  without  action  unless  the 
molecules  of  the  two  gases  are  mixed.  It  has  recently  been 
shown  by  Luther  and  Goldberg,  t  that  this  behavior  of  a  mixture 
of  hydrogen  and  chlorine  is  probably  due  to  the  presence  of  a 
minute  quantity  of  oxygen  which  acts  as  a  negative  catalyst  to 
the  reaction.  By  allowing  the  light  from  a  constant  source  of 
illumination  to  pass  through  a  layer  of  chlorine  of  definite  thick- 
ness, Bunsen  and  Roscoe  determined  how  much  energy  was 
absorbed.  Then  after  repeating  the  experiment  with  a  layer  of 

*  Ann.  Chim.  Phys.  [3],  9,  257  (1843). 
t  Pogg.  Ann.,  100,  481  (1857). 
J  Zeit.  phys.  Chem.,  56,  43  (1906). 


450  THEORETICAL  CHEMISTRY 

hydrogen  of  the  same  thickness,  they  determined  the  absorption 
of  a  mixture  of  equal  volumes  of  hydrogen  and  chlorine,  the  thick- 
ness of  the  absorbing  layer  being  double  that  of  either  gas  alone. 
The  absorption  due  to  the  mixture  of  the  two  gases  was  greater 
than  the  sum  of  the  absorptive  effects  of  the  components  taken 
separately.  This  difference  in  absorbing  power  Bunsen  and 
Roscoe  considered  to  be  a  measure  of  the  amount  of  radiant  energy 
required  to  cause  the  formation  of  hydrochloric  acid  from  its  con- 
stituent elements.  They  found  that  approximately  two-thirds  of 
the  radiant  energy  absorbed  by  the  mixture  appeared  in  the  form 
of  heat,  while  the  remaining  one-third  was  required  to  initiate  the 
reaction. 

Photochemical  Extinction.  As  has  already  been  stated,  only 
that  fraction  of  the  radiant  energy  which  is  absorbed  by  a  sub- 
stance is  capable  of  bringing  about  chemical  change.  Bunsen 
and  Roscoe  discovered  that  when  light  is  allowed  to  pass  through 
a  layer  of  hydrogen  and  chlorine  there  is  a  loss  in  photochemical 
activity,  so  that  if  the  beam  of  light  after  traversing  the  first  layer 
of  hydrogen  and  chlorine  be  allowed  to  pass  through  a  second 
similar  layer,  there  will  be  a  correspondingly  smaller  amount  of 
hydrochloric  acid  formed.  This  phenomenon  they  termed  photo- 
chemical extinction.  As  a  result  of  their  investigations,  Bunsen 
and  Roscoe  concluded  that  a  definite  photochemical  change  is 
produced  by  a  definite  amount  of  radiant  energy.  This  general- 
ization, sometimes  referred  to  as  the  first  law  of  actinochemistry, 
may  be  stated  as  follows:  The  amount  of  photochemical  action  is 
equal  to  the  product  of  the  intensity  of  the  light  and  the  time  during 
which  it  acts. 

Some  Photochemical  Reactions.  As  is  well  known,  the 
science  of  photography  is  based  upon  the  action  of  light  on  the 
halogen  salts  of  silver,  the  so-called  latent  image  being  formed  by 
photochemical  reduction.*  White  phosphorous  is  slowly  changed 
into  the  red  modification  when  exposed  to  sun-light.  This 

*  For  an  excellent  treatment  of  the  chemistry  of  photography  the  student 
is  recommended  to  consult,  "Photography  for  Students  of  Physics  and  Chem- 
istry," by  Louis  Derr,  Macmillan;  or  "Photochemie  und  Beschreibung  der 
photographischen  Chemikalien, "  by  H.  W.  Vogel. 


ACTINOCHEMISTRY  451 

change  seems  to  be  independent  of  the  surrounding  medium, 
taking  place  equally  well  whether  the  element  be  immersed  in 
water  or  ether,  or  in  an  atmosphere  of  hydrogen.  The  decom- 
position of  aqueous  solutions  of  hydrogen  peroxide  and  of  chlorine, 
and  the  reduction  of  potassium  dichromate  by  gelatine  are  familiar 
examples  of  photochemical  action.  When  bromine  and  toluene 
react  in  the  dark  or  in  diffuse  light,  a  mixture  of  ortho-  and  para- 
bromtoluene  is  obtained,  whereas  if  the  reacting  substances  are 
brought  together  in  bright  sun-light  an  isomer,  benzyl  bromide  is 
formed.  It  has  been  shown  that  a  number  of  organic  acids 
undergo  transformation  into  stereoisomeric  substances  on  expo- 
sure to  light,  the  transformation  being  accompanied  by  an  evolution 
of  heat.  In  general,  it  may  be  said  that  radiant  energy  is  capable 
of  causing  oxidation,  reduction,  polymerization,  and  decomposition. 
While  there  is  no  general  rule  governing  the  action  of  light,  it  has 
been  shown  in  a  large  number  of  cases  that  metallic  compounds 
are  oxidized  by  long  wave-lengths  and  reduced  by  short  wave- 
lengths. 

Emission  of  Light.  Having  considered  very  briefly  the 
transformation  of  radiant  energy  into  chemical  energy,  the  re- 
mainder of  this  chapter  will  be  devoted  to  an  equally  brief  treat- 
ment of  the  reverse  process,  —  the  transformation  of  chemical 
energy  into  radiant  energy.  With  the  exception  of  the  electric 
light,  all  sources  of  artificial  illumination  are  dependent  upon  the 
heat  developed  by  chemical  reactions.  It  is  a  familiar  fact  that  all 
solids  and  liquids  when  heated  to  a  sufficiently  high  temperature 
emit  light.  At  a  temperature  of  about  525°  they  all  begin  to 
glow  with  a  dull  red  light  and  as  the  temperature  is  raised  the  light 
emitted  is  found  to  contain  an  increasing  number  of  shorter  wave- 
lengths, until  eventually  at  a  white-heat  a  continuous  spectrum 
is  obtained.  Before  stating  the  fundamental  laws  of  thermal 
radiation  it  may  be  well  to  define  several  terms  with  which  the 
student  may  be  more  or  less  familiar. 

A  black  body  is  one  which  absorbs  completely  all  of  the  incident 
radiation.  By  the  absorbing  power  of  a  body  is  to  be  understood 
the  fraction  of  the  incident  radiation  which  it  absorbs,  and  by  the 
term,  emissivity,  is  meant  the  ratio  of  the  amount  of  energy  radi- 


452  THEORETICAL  CHEMISTRY 

ated  to  the  energy  which  would  be  radiated  by  a  similar  black 
body. 

Kirchhoff's  Law.  In  1857,  Kirchhoff  discovered  an  important 
relation  between  emissive  and  absorptive  power,  commonly  known 
as  Kirchhoff's  law.  This  may  be  stated  as  follows:  At  a  given 
temperature  the  ratio  between  the  emissive  and  absorbing  powers,  for 
a  definite  wave-length,  is  the  same  for  all  bodies.  Hence,  since  a 
black  body  has  maximum  absorbing  power  it  will  also  have 
maximum  emissive  power. 

A  close  approximation  to  a  perfectly  black  body  is  furnished 
by  a  hollow  sphere  furnished  with  a  small  window  through  which 
observations  can  be  made.  Any  radiations  which  enter  through 
the  window  are  either  absorbed  or  suffer  multiple  reflection 
without  emergence. 

Stefan's  Law.  It  was  shown  by  Stefan,*  and  subsequently 
by  Boltzmann,f  that  the  total  radiation  from  a  perfectly  black  body 
is  proportional  to  the  fourth  power  of  the  absolute  temperature;  or 
if  S  is  the  complete  emission  of  a  black  body,  and  T  is  its  absolute 
temperature, 


where  a  is  a  constant.     This  law  has  been  experimentally  verified 
by  Kurlbaum,t  who  found  that 

a  =  1.71  X  10~5  ergs  per  sec.  =  0.408  X  10~12  calories  per  sec. 

Wien's  Law  of  Displacement.  Another  important  law  of 
radiation,  discovered  by  Wien,§  may  be  stated  as  follows:  The 
wave-length  corresponding  to  the  maximum  emission  of  a  perfectly 
black  body  is  inversely  proportional  to  its  absolute  temperature. 

If  Xm  denotes  the  wave-length  corresponding  to  maximum  radi- 
ation, and  T  is  the  absolute  temperature,  Wien's  law  may  be  ex- 
pressed mathematically  as  follows  :  — 

\mT  =  constant  =  k. 

*  Ber.  d.  k.  Akad.,  Wien,  79,  391  (1879). 
t  Wied.  Ann.,  22,  291  (1884). 
j  Wied.  Ann.,  65,  746  (1898). 
§  Ibid.,  52,  32  (1894). 


ACTINOCHEMISTRY 


453 


The  mean  value  of  k  for  a  perfectly  black  body  is  2940,  when  Xm 
is  expressed  in  microns.  This  law  has  also  been  verified  experi- 
mentally. Some  of  the  results  obtained  in  this  direction  by 


Wave-length  r 
Fig.  104. 

Lummer  and  Pringsheim  *  are  shown  in  the  accompanying  dia- 
gram Fig.  104.  The  ordinates  are  proportional  to  the  energy  of 
the  radiations,  while  wave-lengths,  expressed  in  microns,  are 

*  Verhand.  Deutsch.  Phys.  Gesel.,  i,  218  (1899). 


454  THEORETICAL  CHEMISTRY 

plotted  as  abscissae.  The  absolute  temperature  of  the  radiating 
body  is  given  with  each  curve.  It  will  be  noticed  that  the  total 
energy,  represented  by  the  area  included  under  each  curve,  in- 
creases, according  to  Stefan's  law,  as  the  temperature  is  raised. 
It  will  also  be  noticed  that  the  wave-length  Xm  at  which  the  energy 
attains  its  maximum  value,  decreases  as  the  temperature  increases. 
Radiation  Pyrometers.  Various  types  of  pyrometers  for  the 
measurement  of  high  temperatures  have  been  based  upon  the 
foregoing  principles.  The  most  accurate  optical  pyrometer  is 
that  invented  by  Holborn  and  Kurlbaum.*  In  this  instrument 
the  current  through  a  small  incandescent  lamp  is  varied  until  its 
light  is  just  eclipsed  by  that  from  the  hot  body.  When  this  point 
of  balance  has  been  reached,  the  incandescent  filament  and  the 
hot  body  have  the  same  temperature.  The  pyrometer  is  cali- 
brated by  determining  the  current  necessary  to  raise  the  filament 
of  the  lamp  to  the  temperature  of  the  standard  black  body,  the 
temperature  of  the  latter  being  determined  by  means  of  a  thermo- 
couple. Of  course,  when  the  instrument  is  used  to  determine  the 
temperature  of  sources  of  radiant  energy  which  differ  widely  in 
character  from  that  of  a  perfectly  black  body,  the  accuracy  of 
the  measurements  is  lessened,  but  in  an  extreme  case,  such  as  that 
presented  by  polished  platinum  at  950°,  the  error  does  not  exceed 

74°.f 

Luminescence.  When  certain  substances  are  subjected  to  the 
action  of  light,  a  secondary  radiation  of  the  shorter  wave-lengths 
is  emitted,  which  is  greatly  in  excess  of  the  normal  amount  corre- 
sponding to  the  temperature  of  the  substance.  This  phenomenon 
is  called  luminescence.  Luminescence  is  thought  to  be  the  result 
of  very  slow  chemical  action.  For  example,  phosphorus  under- 
goes slow  oxidation  on  exposure  to  the  air  at  a  low  temperature, 
and  emits  a  pale  white  light.  This  action  is  commonly  called 
phosphorescence  although  the  term  chemi-luminescence  is  to  be 
preferred.  The  term  phosphorescence  more  strictly  applies  to 

*  Ber.  Akad.  d.  Wiss.,  Berlin  (1901),  p.  712. 

t  For  detailed  descriptions  of  different  types  of  pyrometers,  the  reader  is 
recommended  to  consult  Le  Chatelier's  "High  Temperature  Measurements." 
Trans,  by  Burgess. 


ACTINOCHEMISTRY  455 

the  glowing  in  the  dark  of  such  substances  as  calcium  sulphide 
after  exposure  to  intense  radiation.  The  light  emitted  by  phos- 
phorescent substances  is  generally  of  greater  wave-length  than 
that  of  the  exciting  radiation.  Certain  substances,  such  as  qui- 
nine, kerosene,  uranium  glass,  and  resorcinolphthalein  emit  light 
of  medium  wave-length  when  exposed  to  ultra-violet  light.  Such 
substances  are  said  to  be  fluorescent. 


INDEX   OF   NAMES 


Abegg,  196. 

Alexieeff,  150,  152. 

Amagat,  49,  50. 

Andrews,  82,  83. 

Aristotle,  2,  6. 

Arrhenius,   189,  205,  206,  207,  318, 

342,  363,  375,  378,  381. 
Aston,  369. 
Avogadro,  9,  10,  17,  51,  54,  55,  63, 

205. 

Babo,  186. 

Baly,  115,  116,  117. 

Bancroft,  380,  423. 

Barlow,  138,  139. 

Bassett,  345. 

Beccaria,  335. 

Bechhold,  230. 

Beckmann,  198. 

Becquerel,  448,  449. 

Beer,  116. 

Bergmann,  262. 

Berkeley,  Earl  of,  175,  178,  179,  214. 

Bernoulli,  51. 

Berthelot,   239,   252,   258,   262,   263, 

264,  273,  276. 
BerthoUet,  4,  5. 
Berzelius,  9,  15,  16,  17,  336. 
Bigelow,  331. 
Biltz,  229,  379. 
Bingham,  94. 
Biot,  107. 
Blagden,  195. 
Bodenstein,  267,  268,  269. 
Boltzmann,  51,  452. 
Boettger,  365. 


Boyle,  Robert,  2,  3,  6,  47,  51,  53,  56, 

57,  73,  80,  170,  175,  236. 
Brauner,  32. 

Bredig,  234,  235,  331,  359. 
Brown,  226. 
Bruhl,  102. 
Budde,  56. 
Buff,  96. 

Bunsen,  147,  148,  446,  447,  449,  450. 
Burnett,  447. 
Burton,  223,  225. 

Cailletet,  88. 

Callendar,  179. 

Cannizzaro,  17. 

Carlyle,  335. 

Charpy,  303. 

Ciamician,  445. 

Clark,  407. 

Claude,  90. 

Clausius,  51,  342,  363. 

Clement,  71,  332. 

Crookes,  Sir  William,  27,  35,  36. 

Dale,  101. 

Dalton,  1,  5,  7,  16,  17,  146,  149,  285. 

Davy,  335. 

Debray,  278. 

De  Chancourtois,  21,  22,  28. 

De  Forcrand,  284,  285. 

Democritus,  6. 

Desch,  117. 

Desormes,  71,  332. 

Deville,  63,  271. 

De  Vries,  179,  180,  181,  182,  208. 

Dewar,  89,  90. 


457 


458 


INDEX  OF  NAMES 


Dobereiner,  21. 

Drude,  126. 

Duclaux,  221. 

Dulong,  11,  12,  13,  14,  29,  139. 

Dumas,  20. 

Dutoit,  367. 

Emerson,  27.  t 

Eotvos,  121. 
Epicurus,  6. 

Fanjung,  368. 

Faraday,  7,  35,  87,  88,  113,  125,  339. 

Faure,  429. 

Fick,  184. 

Forbes,  416. 

Fraunhofer,  114. 

Freudenberg,  442. 

Freundlich,  227,  232. 

Friderich,  369. 

Fuchs,  433. 

Gay-Lussac,  8,  9,  47,  54,   170,   174, 

175,  186. 
Geoffrey,  262. 
Getman,  120,  212. 
Gibbs,  288,  397. 
Gladstone,  101. 
Goldberg,  449. 
Goodwin,  418. 

Graham,  55,  184,  219,  220,  233. 
Grotthuss,  341. 
Grove,  342. 

Guldberg,  5,  94,  264,  279. 
Gutbier,  234. 
Guthrie,  299,  306. 
Guye,  94,  113. 

Hamburger,  182,  183. 
Hampson,  89,  90. 
Hardy,  223,  228. 
Hartley,  115,  117,  119,  175. 
Hautefeuille,  267. 
Haiiy,  138. 
Heycock,  303. 


Hedin,  183. 

Helmholtz,  397,  401,  412. 
Henry,  110,  149,  285,  408. 
Heydweiller,  364. 
Hess,  240,  243,  254. 
Hittorf,  343. 
Holborn,  454. 
Horstmann,  281. 
Hull,  445. 
Hulett,  159. 

Isambert,  281. 

Jones,  197,211,216,345,368. 
Jurin,  119. 

Kirchhoff,  452. 

Knoblauch,  325. 

Kcelichen,  376. 

Kohlrausch,  350,  356,  363,  364,  366. 

Konowalow,  153,  154. 

Kopp,  63,  65,  95,  96,  102,  141. 

Kroenig,  51. 

Kundt,  71,  73. 

Kurlbaum,  452,  454. 

Landolt,  4. 

Langbein,  252. 

Laplace,  73,  240. 

Larmor,  323. 

Lavoisier,  2,  3,  236,  240. 

Lebedew,  445. 

Le  Bel,  109,  110,  112. 

Le  Blanc,  433,  435,  437,  441. 

Le  Chatelier,  70,  259,  260,  445. 

Lemoine,  267. 

Lenard,  39. 

Leucippus,  6. 

Lewis,  139,  140. 

Liebig,  332. 

Linde,  89,  90. 

Linder,  222,  232. 

Lippmann,  408. 

Lodge,  361. 

Longinescu,  142. 


INDEX  OF  NAMES 


459 


Loomis,  196. 
Lorentz,  104. 
Lorenz,  104. 
Lessen,  96. 
Lummer,  453, 
Lunden,  392. 
Luther,  449. 

Marignac,  5,  16. 

Maxwell,  51. 

Mayer,  A.  M.,  45. 

Mayer,  J.,  240. 

Mendeleeff,  3,  17,  22,  23,  28,  29,  30, 

31,  46. 

Menschutkin,  328. 
Meyer,  416. 
Meyer,  Lothar,  22,  28. 
Meyer,  Victor,  58,  59,  61,  62,  69. 
Miller,  131. 

Mitscherlich,  14,  15,  138. 
Morgan,  124,  125. 
Morley,  16,  17,  59. 
Morse,  172,  174,  175,  176,  214. 

Natanson,  66. 

Natterer,  56. 

Naumann,  131,  132. 

Nernst,  94,  116,  126,  185,  186,  260, 

285,  286,  289,  369,  400,  413,  419. 
Neumann,  13,  14,  424. 
Neville,  303. 
Newlands,  22. 
Newton,  7,  262. 
Nichols,  445. 
Nicholson,  335. 
Nollet,  166. 
Noyes,  A.  A.,  322,  327,  367. 

Ohm,  338. 

Olszewski,  88,  89. 

Onnes,  91. 

Ostwald,  W.,  8,   15,   122,   147,   169, 

209,  237,  318,  329,  333,  343,  372, 

374,  384,  396,  448. 
Ostwald,  Wo.,  224. 


Palmaer,  402. 

Pasteur,  107,  108,  109. 

Pean  de  St.  Gilles,  263,  264,  273. 

Pebal,  64,  65. 

Perkin,  W.  H.,  113,  114. 

Perrin,  38,  226. 

Petit,  11,  12,  13,  14,  29,  139. 

Pfeffer,  166,  167,  170,  171,  172,  176. 

Philip,  148,  214. 

Pictet,  88. 

Picton,  222,  232. 

Planck,  185. 

Plante,  429. 

Plato,  6. 

Poggendorff,  404. 

Pope,  138,  139. 

Pringsheim,  453. 

Proust,  4,  5. 

Prout,  16,  17,  20. 

Pulfrich,  99. 

Ramsay,  76,  121,  122,  123,  125,  142, 

143,  146. 

Raoult,  187,  190,  191,  193,  196,  201. 
Regnault,  58. 
Reicher,  318. 

Richards,  16,  17,  139,  416. 
Richter,  4. 
Rigollot,  448. 

Roberts- Austen,  163,  303,  305. 
Roozeboom,  301,  303. 
Roscoe,  155,  446,  447,  449. 
Rose,  263. 
Rudolphi,  380. 

Schiff,  96. 

Senier,  2. 

Shields,  121,  122,  123,  125,  142,  143. 

Siedentopf,  225. 

Snell,  99. 

Soddy,  3. 

Soret,  186. 

Stas,  16,  17,  20, 


460 


INDEX  OF  NAMES 


Steele,  362. 
Stefan,  452. 
Steno,  135. 
Stokes,  42. 
Svedberg,  235. 

Tammann,  184. 

Thilorier,  88. 

Thomsen,  252. 

Thomson,  J.  J.,  38,  39,  43,  44,  45,  46, 

337,  369,  397. 
Thorpe,  96. 
Traube,  97. 
Traube,  M.,  166. 
Trouton,  93,  94. 

Van  der  Stadt,  323. 

Van  der  Waals,  56,  57,  82,  84,  85,  86, 

87,  96,  149. 
Van't  Hoff,  109,  110,  112,  146,  161, 

162,  163,  170,  171,  174,  175,  191, 

195,  196,  201,  204,  267,  274,  288, 

328,  380. 
Volta,  335. 


Waage,  5,  264,  279. 
Walden,  354. 
Walker,  3,  92,  189,  326. 
Warder,  318. 
Washburn,  367. 
Weber,  12,  13. 
Weimarn,  224. 
Weiss,  129. 
Wenzel,  262. 
Weston,  406. 
Whetham,  362. 
Whitney,  327. 
Wien,  452. 
Williamson,  342. 
Wilsmore,  413. 
Wilson,  39. 
Winkler,  30. 
Wladimiroff,  183. 
Wroblewski,  88,  89. 
Wiillner,  186. 

Young,  161. 
Zsigmondy,  225,  230. 


INDEX  OF   SUBJECTS 


Abnormal  solutes,  203. 
Absolute  index  of  refraction,  100. 
Absorption,  photochemical,  448. 

spectra,  114. 
Accumulators,  428. 
Actinochemistry,  444. 
Actinometer,  446. 

electrical,  448. 

hydrogen-chlorine,  446. 

mercuric  oxalate,  447. 

silver  chloride,  447. 
Adsorption,  229,  232. 

equilibrium,  232. 
Alloys,  303. 

eutectic,  303. 
Ampere,  338. 
Anion,  340. 
Anode,  340. 
Association,  203. 

in  solution,  203. 
Atomic  theory,  6,  7. 

weight,  11,  16,  18. 

weight,  correction  of,  31. 

weight,  estimation  of,  29. 
Autocatalysis,  331. 

Basicity  of  organic  acids,  384. 
Bimolecular  reactions,  316. 
Boiling-point  constant,  191. 

and  critical  temperature,  94. 

elevation  of,  191. 

elevation    and    osmotic    pressure, 

194. 
Brownian  movement,  226. 

Calorie,  236. 
Calorimeter,  238. 


461 


Calorimeter,  combustion,  239. 
Capillary  electrometer,  407. 
Carbon  dioxide,  isothermals  of,  80. 
Catalysis,  329. 

mechanism  of,  332. 
Catalyst,  negative,  332. 
Cataphoresis,  223. 
Cathode,  35,  340. 

particle,  charge  carried  by,  41. 

particle,  velocity  of,  39. 

rays,  properties  of,  36. 
Cation,  340. 
Cells,  galvanic,  395. 

gas,  425. 

reversible,  397. 

standard,  406. 

storage,  428. 

Chemical  constitution,  107,  117. 
Chemical  and  electrical  energy,  re- 
lation between,  397. 
Chemical  kinetics,  309. 
Chemical  properties  of  ionized  solu- 
tions, 209. 

Chemi-luminescence,  454. 
Clark  ceU,  407. 
Coagulation  of  suspensoids,  227. 

reciprocal,  229. 
Coefficient,  isotonic,  181. 

diffusion,  184. 

solubility,  149. 
Colloidal  solutions,  219,  220. 

density  of,  220. 

electrical  properties  of,  222. 

magnetic  properties  of,  222. 

osmotic  pressure  of,  220. 

preparation  of,  233. 


462 


INDEX  OF  SUBJECTS 


Colloidal  suspensions,  223. 
Colloids,  219. 

irreversible,  220. 

molecular  weight  of,  222. 

protective,  230. 

reversible,  220. 

surface  energy  of,  231. 
Combustion,  heats  of,  252. 

bomb,  240. 
Compounds,  2,  7. 
Compressibilities   of  solid  elements, 

139. 

Concentration  elements,  415,  417. 
Conductance,  at  high  temperatures, 
367. 

and  ionization,  362. 

determination  of,  350. 

electrical,  335. 

of  difficultly  soluble  salts,  365. 

of  fused  salts,  370. 

of  non-aqueous  solutions,  368. 

relative,  352. 
Conduction    of    electricity    through 

gases,  34. 

Connection    between     gaseous    and 
liquid  states,  79. 

between  lowering  of  vapor  pressure 

and  osmotic  pressure,  189. 
Consecutive  reactions,  326. 
Constant  pressure,  reactions  at,  248. 
Constant  volume,  reactions  at,  248. 
Constants,  dielectric,  125. 
Constitution,  chemical,  107,  117. 
Continuity    of    gaseous    and    liquid 

states,  82. 
Corpuscle,  43. 
Corresponding  conditions,  85,  86. 

temperatures,  86. 
Coulomb,  338. 
Counter  reactions,  325. 
Co-volume,  96. 
Critical  pressure,  80. 

solution  temperature,  152. 

temperature,  80. 


Critical   pressure,    temperature   and 

boiling-point,  94. 
volume,  80. 

Crookes'  dark  space,  35. 

Cryohydrate,  297,  299. 

Crystal  form  and  chemical  composi- 
tion, 138. 

Crystalloids,  219. 

Crystallography,  129. 

Crystals,  properties  of,  136. 

Current,  residual,  434. 

Decomposition  potential,  432. 
Deductions  from  kinetic  equation,  53. 
Degree  of  dissociation,  66. 

of  freedom,  289. 

Density  of  colloidal  solutions,  220. 
'  Derivation  of  kinetic  equation,  52 
Dialysis,  219. 
Dialyzer,  219. 
Dielectric  constant,  125. 
Diffusion,  coefficient  of,  184. 

and  osmotic  pressure,  184. 
Dilution,  heat  of,  247. 
Dilution  law,  372. 
Dimorphism,  138. 
Dispersoids,  224. 
Dissociation,  203. 

constant,  270. 

hydrolytic,  387. 

in  solution,  204. 
Distillation,  fractional,  156. 

steam,  157.  4 

Distribution  of  solute  between  immis- 
cible solvents,  285. 

Electrical  charge  and  mass,  43. 
Electrical  conductance,  335. 

double  layer,  401. 
Electrodes,  dropping,  412. 

hydrogen,  413. 

of  first  type,  418. 
Electrodes,  normal,  410. 

null,  412. 


INDEX  OF  SUBJECTS 


463 


Electrodes,  of  second  type,  418. 

unpolarizable,  435. 
Electrolysis,  340,  432. 
Electrolytes,  ionization  of  strong,  379. 

mixtures  with  a  common  ion,  377. 
Electrolytic  dissociation,  205. 

separation  of  metals,  442. 

solution  pressure,  423. 
Electrometer,  capillary,  407. 
Electromotive  force,  395. 

measurement  of,  404. 
Electron,  43. 

hypothesis,  44. 

theory,  34. 
Elements,  classification  of,  20,  29. 

oxidation  and  reduction,  421. 

periodic  series  of,  24. 
Elevation  of  boiling-point,  191. 
Emulsoids,  226. 

precipitation  of,  229. 
Equation  of  condition,  86. 

of  Van  der  Waals,  56,  82. 
Equilibrium,  adsorption,  232. 

constant,  266. 

heterogeneous,  278. 

homogeneous,  262. 

in  homogeneous  gaseous  systems, 
267. 

in  liquid  systems,  272. 

variation  with   temperature,   274, 

287. 
Equivalent,  chemical,  6. 

electrochemical,  340. 
Etch  figures,  137. 
Extinction,  photochemical,  450. 

Faraday,  338. 
Ferments,  inorganic,  331. 
Fluorescence,  455. 
Formation,  heat  of,  244. 
Fractional  distillation,  156. 
Freezing-point,    depression    and    os- 
motic pressure,  199,  211. 
lowering  of,  195. 


Fundamental  principles,  1. 
Fusion,  heat  of,  128. 

Galvanic  cell,  395. 
Gas  cells,  425. 

laws,  deviations  from,  49. 
Gases,  47. 

kinetic  theory  of,  51. 

liquefaction  of,  87. 

solutions  of  gases  in,  145. 

solutions  of  in  liquids,  147. 
Goniometer,  135. 

Haematocrit,  183. 

Heat,  molecular  specific,  71. 

of  combustion,  252. 

of  dilution,  247. 

of  dissociation  of  solids,  284. 

of  formation,  244. 

of  fusion,  128. 

of  hydration,  247. 

of  ionization,  257,  381,  424. 

of  neutralization,  255. 

of   reaction,    variation   with   tem- 
perature, 250. 

of  solution,  245. 

of  vaporization,  93. 

specific,  11,  67. 

specific,  at  constant  pressure  and 
volume,  68. 

specific,  of  gases,  75. 

specific,  of  solid  elements,  139. 
Helix  of  de  Chancourtois,  21. 
Heterogeneous  equilibria,  law  of  mass 

action  applied  to,  278. 
Heterogeneous  gaseous  systems,  equi- 
librium in,  267. 

Heterogeneous  reactions,  velocity  of, 
326. 

systems,  278. 
Hydrogel,  229. 
Hydrolysis,  386. 

determination  of,  391, 


464 


INDEX  OF  SUBJECTS 


Hypothesis,  2. 
of  Avogadro,  9. 
of  Prout,  20. 

Immiscibility,  156. 
Index  of  refraction,  99. 
Induction,  period  of,  449. 

photochemical,  449. 
Internal  compensation,  112. 
Ionic  product,  383. 
Ionic  velocity,  absolute,  359. 
lonization  and  conductance,  362. 
lonization  constant,  379. 

heat  of,  257,  381,  424. 

influence  of  substitution  on,  385. 

of  strong  electrolytes,  379. 

of  water,  427. 

Ionized  solutions,  properties  of,  209. 
Ionizing  power  of  solvents,  369. 
Ions,  205. 

absolute  velocity  of,  359. 

existence  of  free,  341. 

migration  of,  343. 
Irreversible  colloids,  220. 
Isohydric  solutions,  378. 
Isomorphism,  14. 
Isothermal,  80. 
Isotonic  coefficient,  181. 

Kinetic  equation,   deductions  from, 

53. 

derivation  of,  52. 
Kinetic  theory  of  gases,  51,  75. 
Kinetics,  chemical,  309. 

Labile  solutions,  161. 
Law  of  Avogadro,  54. 

of  Beer,  116. 

of  Boyle,  47,  53. 

of  combining  proportions,  5. 

of  conservation  of  mass,  3. 

of  Dalton,  5. 

of  definite  proportions,  4. 

of  Dulong  and  Petit,  11. 


Law  of  Faraday,  339. 

of  Gay-Lussac,  8,  47,  54. 

of  Graham,  55. 

of  Guldberg  and  Waage,  264. 

of  Henry,  148. 

of  Hess,  254. 

of  Jurin,  119. 

of  Kirchhoff,  452. 

of  Kohlrausch,  356. 

of  Lavoisier  and  Laplace,  240. 

of  mass  action,  264. 

of  Mitscherelich,  14. 

of  multiple  proportions,  5. 

of  Neumann,  14. 

of  octaves,  22. 

of  Ohm,  338. 

of  Raoult,  187. 

of  Richter,  4. 

of  Stefan,  452. 

of  Wien,  452. 

periodic,  22. 
Light,  emission  of,  451. 
Liquefaction  of  gases,  87. 
Liquids,  79. 

general  characteristics  of,  79. 
•     refractive  power  of,  98. 

solution  of  liquids  in,  150. 

vapor  pressure,  92. 
Liquid  systems,  equilibrium  in,  272. 
Luminescence,  454. 

Magnetic  rotation,  113. 
Mass  action,  law  of,  264. 
Maximum  work,  principle  of,  258. 
Mean  free  path,  51. 

velocity  of  gaseous  molecule,  55. 
Measurement    of    osmotic    pressure, 

167. 

Mechanism  of  catalysis,  332. 
Melting-point,  129. 
Membranes,  semi-permeable,  165. 
Metastable  solutions,  161. 
Miscibility,  complete,  152. 

partial,  150. 


INDEX  OF   SUBJECTS 


465 


Molecular  gas   constant,   evaluation 

of,  49. 

magnetic  rotation,  114. 
refraction,  101. 
rotation,  106. 
vibration,  117. 
volume,  95. 
weight,  58. 
weight   by   boiling-point   method, 

193. 
weight  by  freezing-point  method, 

198. 

weight  of  colloids,  222. 
weight  of  solids,  142. 
weight  in  solution,  200. 
weight  from  surface  tension,  121. 

Negative  glow,  35. 
Neutralization,  255. 
Non-dissociating  solvent,  solution  of 

solid  in,  286. 
Normal  electrodes,  410. 

Occlusion,  163. 
Octaves,  law  of,  22. 
Ohm,  335. 

Optical  activity,  107. 
Osmotic  pressure,  165. 

and  boiling-point  elevation,  194. 
and  diffusion,  184. 
and  freezing-point  depression,  199. 
and   lowering   of   vapor   pressure, 

189.   . 

measurement  of,  167,  172. 
of  colloidal  solutions,  220. 
Osmotic  pressure  and  nature  of  mem- 
brane, 169. 

theoretical  value  of,  170. 
Osmotic    pressures,    comparison    of, 

179. 

Oxidation   and   reduction   elements, 
421. 

Periodic  law,  22. 
applications  of,  29. 


Periodic  law,  defects  in,  32. 

graphic  representations  of,  27. 
Phase  rule,  288. 

derivation  of,  289. 
Phosphorescence,  454. 
Photochemical  absorption,  448. 

extinction,  449. 

induction,  449. 

reactions,  450. 
Photochemistry,  444. 
Physical  properties  of  ionized  solu- 
tions, 209. 
Plasmolysis,  180. 
Plasmolytic  methods,  179. 
Polarization,  432. 

capacity,  434. 

electromotive  force  of,  432. 

theory  of,  437. 
Polarized  light,  rotation  of  plane  of, 

104. 

Polymorphism,  138. 
Positive  column,  35. 
Potential  between   metal  and  solu- 
tion, 403. 

decomposition,  432,  435. 

difference  at  junction  of  two  solu- 
tions, 419. 

measurement  of  difference  between 

metal  and  solution,  414. 
Precipitation  of  emulsoids,  229. 
Preparation    of    colloidal    solutions, 

233. 
Principle  of  maximum  work,  258. 

of  Soret,  186. 
Properties,  of  cathode  rays,  36. 

electrical  and  magnetic,  of  colloidal 
solutions,  222. 

of  crystals,  136. 

of  solids,  128. 

periodicity  of,  28. 
Proportions,  law  of  combining,  5. 

law  of  definite,  4. 

law  of  multiple,  5. 
Protective  colloids,  230. 


466 


INDEX  OF  SUBJECTS 


Protoelements,  32. 
Pyrometers,  radiation,  454. 

Radiant  energy,  444. 
Radiation  pyrometers,  454. 
Ratio  of  charge  to  mass,  41. 

of  specific  heats,  71. 

of  specific  heats,  determination  of, 

71." 

Reaction,  determination  of  order  of, 
324. 

isochore,  274. 

isotherm,  267. 

velocities,  complex,  324. 

velocity,  influence  of  solvent  on, 
328. 

velocity  and  temperature,  327. 
Reactions,  consecutive,  326. 

at  constant  pressure,  248. 

at  constant  volume,  248. 

counter,  325. 

heterogeneous,  velocity  of,  326. 

of  first  order,  311,  314. 

of  higher  orders,  323. 

of  second  order,  316. 

of  third  order.  320. 

photochemical,  450. 

side,  325. 

Reciprocal  coagulation,  229. 
Refraction,  index  of,  99. 

molecular,  101. 
Refraction,  specific,  101,  103. 
Residual  current,  434. 
Resistance  capacity,  351. 
Reversible  cells,  397. 

colloids,  220. 
Rotation,  magnetic,  113. 

molecular,  106. 

of  plane  polarized  light,  104. 

specific,  106. 

Salt   solutions,  thermoneutrality   of, 

254. 
Saturated  solution,  147. 


Semi-permeable  membranes,  165. 

Side  reactions,  325. 

Solid  solutions,  162. 

Solids,  general  properties  of,  128. 

heat  of  dissociation  of,  284. 

molecular  volumes  of,  141. 

molecular  weights  of,  142. 
Solubility,  coefficient  of,  149. 
Solubility  product,  382. 
Solution,  association  in,  203. 

dissociation  in,  204. 

heat  of,  245. 

of  solid  in  non-dissociating  solvent, 
286. 

pressure,  399. 
Solutions,  145. 

classification  of,  145.  9 

colloidal,  219,  220. 

conductance  of  non-aqueous,  '368. 

of  gases  in  gases,  145. 

of  gases  in  liquids,  147. 

of  solids  in  liquids,  159. 

isohydric,  378. 

labile,  161. 

of  liquids  in  liquids,  150. 

metastable,  161. 

saturated,  159. 

solid,  162. 

supersaturated,  159. 

volume-normal,  174. 

weight-normal,  174. 
Solvate  theory,  214. 
Solvates,  214. 
Specific  heat,  11,  67. 

at  constant  pressure  and  volume, 
68. 

of  gases,  75. 

of  solid  elements,  139. 
Specific  refraction,  101,  103. 
Specific  rotation,  106. 
Spectograph,  115. 
Spectra,  114. 
Standard  cells,  406,  407. 
Steam  distillation,  157. 


INDEX  OF  SUBJECTS 


467 


Strength  of  acids  and  bases,  374. 

Sublimation,  128. 

Sulphur,  equilibrium  between  phases 

of,  293. 

Surface  energy  of  colloids,  231. 
Surface  tension,  119. 

and  drop- weight,  124. 

and  molecular  weight,  121. 
Suspensoids,  226. 

coagulation  of,  227. 
Systems,  three-component,  306. 

two-component,  295. 

Thermal  units,  236. 
Theorem  of  Berthelot,  258. 

of  Le  Chatelier,  259. 
Theory,  2. 

electron,  21. 

of  electrolytic  dissociation,  205. 

kinetic,  75. 

solvate,  214. 
Thermochemical  equations,  237. 

measurements,  238. 
Thermochemistry,  236. 
Thermoneutrality   of   salt   solutions, 

254. 

Thilorier's  mixture,  88. 
Transition  point,  293. 
Transport  numbers,  345. 

determination  of,  345. 
Triads,  Dobereiner's,  21. 
Trimolecular  reactions,  320. 
Triple  point,  292. 
Tyndall  phenomenon,  225. 


Ultramicroscope,  225. 
Unimolecular  reactions,  311. 
Units,  electrical,  338. 

Valence,  15. 
Vapor  density,  58. 

abnormal,  63. 

determinations,  61. 
Vaporization,  heat  of,  73. 
Vapor  pressure,  92. 

connection    between    lowering    of 
and  osmotic  pressure,.  189. 

lowering  of,  186. 

of  liquids,  92. 

Variation    of     equilibrium    constant 
with  temperature,  274,  287. 

of  heat  of  reaction  with  tempera- 
ture, 250. 

Velocity  of  gaseous  molecule,  55. 
Vibration,  curves  of  molecular,  117. 

molecular,  117. 
Volt,  338. 
Voltaic  pile,  335. 
Volume,  molecular,  95. 

Water,  dissociation  of,  364. 

equilibrium  between  phases  of,  290. 

ionization  constant  of,  392. 

ionization  of,  427. 

primary  decomposition  of,  441. 
Watt-second,  339. 
Weight,  atomic,  8,  10,  17. 

combining  5,  8. 

molecular,  10,  17,  58. 
Weston  cell,  406. 


UNIVERSITY  OF  CALIFORNIA  LIBRARY 
BERKELEY 

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